The large heterogeneity of individuals in their sexual behaviour is perhaps most apparent in the distribution of the number of partners that individuals in a population have had partnerships with. This lifetime number of partnerships ranges from 1 to 600–1000 partners for sexually active individuals [11], [19], [20], and is commonly presented as a cumulative lifetime number of partnerships (CLNP) plot. One of the remarkable features of the CLNP is that the higher end of its distribution ( partners) has a powerlaw-like distribution [19] (Fig. 1).

Whether simulation models can accurately reproduce the heterogeneity in the lifetime number of partners depends on their implementation of the “core group”: a label introduced by Yorke et. al. in 1978 [23], [24] for sexually highly active individuals that have many, possibly concurrent partnerships in a short period of time. Many models struggle with reproducing the powerlaw-like distribution of the CLNP, and typically end up with a distribution that reflects the two or three levels of sexual activity defined in these models (e.g., moderate, intermediate, and core-group individuals [7], [9]–[11]).

We concluded that behavioural heterogeneity in sexually highly active individuals is underestimated in models that distinguish a limited number of sexual activity levels, even if there is stochastic variation among individuals. Rather than using a limited number of sexual activity levels, we correlated the capacity for concurrent partnerships in core-group individuals with two pre-determined characteristics of high risk behaviour, namely the onset and the duration of the core-group period for an individual (Table 1). The earlier the onset, and the longer the duration, the higher would be their maximum number of concurrent partnerships (see methods for details). Furthermore, we controlled the total number of partnerships within the moderate group and within the core-group separately (Table 2). This allowed us to independently regulate how close the number of partnerships within each group would be to the maximum number of partnerships that that group could maintain (i.e. the sum of their capacities). For example, a high number of partnerships in the core-group, relative to the sum of their capacities would result in high lifetime number of partnerships per core-group member without affecting the rate at which moderate individuals acquired new partners. These adjustments gave us considerable control on the shape of the CLNP plot (Fig. 1), and resulted in a powerlaw-like distribution of the CLNP that is similar to that of real sexual contact networks [11], [19], [20].

The current model accurately replicates the CLNP observed in real sexual contact networks. However, similar CLNP distributions can be the result of very different age distributions at which the individuals acquire most of their lifetime number of partners [17]. This was recently demonstrated in a comparison study of three models of sexual contact networks [11]. Because the age at which individuals are exposed to many sexual partners is also the age at which they are most vulnerable for contracting, as well as most efficient at transmitting STIs [11], [22], it is necessary for sexual network models to accurately simulate the distribution of recent number of partners for different age groups.

In the model described by Kretzschmar et. al. in 1996 [7], all individuals were available for sexual partnership from the moment they entered the population at age 15, and 5% of these individuals were labeled as core-group members, and will remain so until the age of 35. The homogeneous age of sexual debut and onset of core-group behaviour of individuals results in a premature age of sexual debut (Fig. 2, gray line), and an overestimation of the mean number of recent partners in the population younger than 35 (Fig. 3 gray line). By adding heterogeneity in the age at which individuals become available for sexual partnerships, and (if applicable) in the onset and duration of their core-group period (Table 1), the current model could accurately match the observed mean number of recent partners per age-group (Fig. 3 orange line).

As the level of sexual activity is such an important indicator of STD risk, we further stratified the recent sexual activity of the Dutch population into the fraction of the population with 1+, 2+, 3+ or 6+ partners in the last half year, as well as by age and gender (Fig. 4). We found that the current model could match many of the features of this more detailed measure of recent sexual activity of the Dutch population. Furthermore, at this level of detail, the recent number of sexual partners appears to be sensitive to most of the model parameters. It can therefore serve as an excellent measure to validate a sexual network model on, but at the same time is difficult to interpret how it depends on these model parameters. What follows is a discussion of how different parts of Fig. 4 relate to the underlying model mechanics.

The fit of the model for individuals with 1+ partnership is predominantly determined by the difference between the total number of partnerships that the model maintains in the moderate group of individuals (Table 2), and the maximum number of partnerships that this group can maintain, given their capacity. If the difference between the two is small individuals will quickly find partners, and few individuals will have been without partner in the last 6 months. A second important factor that influences the fit for individuals with 1+ as well as for those with 2+ partnerships is the duration and relative frequency of different partnership types. The model defines three types of partnerships (short-, medium-, and long-term), all of which have exponentially distributed durations, but with different average lengths of a partnership (see next section, and methods). If, for example, the relative frequency of short-term partnerships is increased, more of the moderate individuals can acquire 1, 2 or even 3+ partnerships in 6 months. Finally, transitional concurrency [25] is a third factor that influences the recent number of partners of moderate individuals. Transitional concurrency is the overlap between the end of one partnership and the beginning of the next. Transitional concurrency is implemented in the model by defining the cost of a partnership as 0 as it nears the end of its duration, thus freeing up “capacity” for the individual (see section on gap length, and methods). If transitional concurrency is set to become possible earlier during partnerships, it will increase the number of moderate individuals that have had 2+ recent partner, but also increases the number of moderate individuals that had no partnerships in the last 6 months.

The group of 3+ recent partnerships (Fig. 4 green lines) past age 30 is predominantly formed by a fraction of the former core-group members that continues to keep an increased capacity () after their main period of core-group behaviour (Table 1). Many moderate individuals will by chance be locked into a long-term partnership at that age, and thus have few opportunities to accumulate recent partners. The group with 6+ recent partnerships is almost exclusively defined by the characteristics of the core-group, i.e. its overall size, the age at which core-group behaviour starts, its duration, and how many partnerships the model maintains within the core-group, compares to the sum of the capacities of that group (Table 1,2).

Partnership duration has been recorded in limited detail and only for the current partner in the Dutch sexual survey (Fig. 5), and thus predominantly reflects the duration of long-term partnerships in the Netherlands. Therefore, we also use the more detailed Natsal 2000 UK survey data [11], [26] to fit the model, which also includes information on the duration of previous partnerships. From the UK survey data it appears that there are three typical durations of partnerships (short-, medium- and long-term, Fig. 5 dashed black line), each of which is exponentially distributed (Table 2).

As mentioned in the previous section, partnership duration (or put more precisely, the relative frequency of long-term partnerships) predominantly affects the number of recent partners of the moderate population; a high relative frequency of long-term partnerships means that many moderate individuals will have had only 1 sexual contact in the last 6 months. Partnership duration has less effect on the number of recent partners of core-group individuals, because core-group members have a capacity of , and are limited to a maximum of 2 concurrent long-term partnerships (see methods). Therefore, core-group members always have capacity available for at least one short or medium-term partnership.

In the model there are no individual sexual behaviour that directly affect the duration of partnerships: durations are determined when a partnership forms, and are independent of the further actions of the individuals involved (such as concurrency). One small exception is that individuals that stop being core-group members have to reduce their current number of concurrent partnerships to match their new maximum capacity (see methods). We conclude that partnership duration shape the sexual network relatively independent from the distribution of the number of partnerships of individuals in the population.

The time between sequential partnerships (e.g. the gap length [21]) is an important factor related to Ct prevalence [12]–[14]: negative gap length (e.g. an overlap in partnerships) signals the amount of concurrency in the population: in contrast to serial monogamy, concurrent partnerships provide a two-way channel for Ct to spread, and not just from a previous partner to a new partner. In contrast, positive gap length indicates the chance that an individual has had to spontaneously clear Ct prior to engaging in a new partnership [13]. The precise definition of gap length is the “time between the end of the second most recent partnership, and the start of the most recent partnership”, in which the order of recentness is determined by the last date that partnerships were still ongoing. Where two or more partnerships are tied for recentness, their order is randomly determined.

The Dutch sexual survey data has insufficient information to reconstruct gap lengths, so the current model was fitted to UK survey data on gap length [11], [20] (Fig. 6), and subsequently adjusted as to fit Dutch survey data on transitional concurrency in recently started ( months) steady partnerships (Fig. 7), where “steady” was interpreted as being either a medium-term or a long-term partnership.

About 3% of the UK population reports a negative gap length of more than 500 days between the last two partnerships (Fig. 6, black line), and up to 35% of the population has had overlap between his/her last two partnerships. These percentages indicate that a model of a sexual contact network should allow concurrency of long-term partnerships (to see overlaps of 500+ days), and should not limit the occurrence of concurrent partnerships to the small core-group of the population, as was the case in the Kretzschmar model. The current model allows transitional concurrency [25] during the last 15% of a partnership for men, and the last 7.5% for women (Fig. 6, orange line). The positive gap length in the UK is relatively short compared to the duration of an untreated Ct infection: only 15% of the population takes more than 1 year to find a new partner.

In the model, the positive gap length is determined by the chance of individuals to participate in the pair-formation process for each timestep (Table 2), and by the total number of partnerships that the model tries to maintain: if either is too low, the time between partnerships (i.e. positive gap size) increases. Positive gap length is also affected by transitional concurrency, as an increase in concurrency increases the fraction of the population that is available for new partnerships, and thus increases the competition for those that are single and seeking a new partnership. The result is that the average positive gap length becomes larger with higher levels of transitional concurrency.

The distribution of Ct prevalence in the population, stratified by level of sexual activity was introduced as a new summary measure by Althaus et. al. [11] that combines sexual behaviour with epidemiological data. As expected, the Ct prevalence within a stratum rises as the level of sexual activity of its individuals rises from 0 to 3 partners in the last year (Fig. 8). Remarkably, however, is that in the UK survey data the Ct prevalence in the subsequent strata (4 and 5+ partners) drops again to the prevalence levels halfway between that of the strata of 2 and 3 partners per year. Neither the current model, nor earlier sexual contact network models [11] are capable of reproducing this observation, or shed light on the mechanism behind it. Two possible mechanisms are explored in Supporting Text S1, namely 1) that prolonged and frequent exposure to Ct could result in a protective immune response [27]–[30], and thus reduce the Ct prevalence in those population groups that are most likely to have experienced a prolonged infection, and 2) that due to coital dilution, that is the decrease in coital frequency when maintaining concurrent partnerships, those with concurrent partnerships will have a reduced chance of acquiring Ct [31]–[33].

A second important feature of the distribution of Ct prevalence in the population is that it provides information about the amount of mixing between moderate and core-group individuals in the population. The level of assortative mixing is an important measure for disease spread [13], [24], but is difficult to measure in surveys, as the surveyed individuals need to have accurate knowledge about the life history of their (possibly short-term) partners. Therefore, models of sexual contact networks typically rely on assumptions on the level of mixing [7], [9], [11]. However, as Ct becomes more concentrated in individuals with a high level of sexual activity [2], [13], [23]

(Fig. 8), in strongly assortative populations, and less so in more well-mixed populations, one can use the difference between the average Ct prevalence of the population, and the Ct prevalence of for example the group of individuals with 1 partnership in the last year, as an observation against which to fit the amount of assortative mixing in the model.

The level of mixing on sexual activity in the model (Table 2) that well describes the UK survey data is a situation in which 87% of the partnerships of the core-group are with moderate members. If we define core-group members as those individuals with 5+ partnerships in the last 12 months, the model results are comparable to earlier investigations on mixing of sexual activity [34], [35], and the amount of mixing in the model can be characterized as having a moderate assortativity coefficient of 0.25 [35].

The population-level summary measures are based on data from the Rutgers Nisso Group Dutch population survey on sexual health in 2009 [40]. This survey entails 6428 individuals that are weighted on gender, age, ethnicity, and the degree of urbanization of their hometown. Individuals that fell outside the age-range of 13–64 were excluded from the data, as were those that reported having had homosexual partnerships, that were paid for partnerships, or had included prostitute visits in their answers on sexual activity. This procedure left a total of 5402 individuals (Table 3). For population-level summary measures which could not be derived from the Dutch sexual survey, we used the Natsal 2000 UK sexual survey [26] as presented in Althaus et. al. [11].

All Dutch survey population measures presented in the manuscript take into account the weighted value of individuals (e.g. individuals that had a weight associated with them because they were sampled from an underrepresented population group also contribute times as much to the population level summary measures as individuals with a weight of 1). The population measure on “sexual activity in the last 6 months” and the model data (Fig. 4) were smoothened to facilitate a visual comparison, using a Savitzky-Golay non-linear smoothing filter [41], [42] (parameters: window size 5, coefficient 4, left-padding 0, right-padding recent mean for 1+ and 2+ partners, and 0 for 3+ and 6+ partners). The smoothing filter works similar to a running average, but performs better at preserving the trends of the survey data.

To measure the Ct distribution in the simulated population, we implemented the Ct infection process as described in Althaus et. al. [11] (Table 4), in which the rate of (unprotected) sexual contacts drops from once every 2 days to once every 7 days after the first two weeks of a partnership. The transmission rate of Ct in our simulations was set to 2.5% per partner per sexual contact, such that the average Ct prevalence in the age-group 18–44 matched the estimated Ct prevalence of 1.7% of the UK in the same age-group [11], [22]. A more in-depth study of the relationship between the decline of condom use and coital frequency during a partnership, and its (limited) effect on the distribution of Ct is presented as Supporting .

The model keeps track of the total number of partnerships between moderate individuals, between core-group members, and between core-group members and moderate individuals (Table 2). Every timestep of the model, any shortage of partnerships in the simulated population is supplemented by randomly sampling pairs of individuals from the population and attempting to form partnerships between them. Once enough new partnerships have been formed, the model moves one timestep ahead and repeats the same process. Because partnerships are continuously dissolved as time progresses, their total number constantly needs to be supplemented.

Not all individuals are available for sampling on a particular day. Individuals have to have the necessary free “capacity” for an additional partnership (taking into account that partnerships that are in their transitional concurrency phase no longer take up capacity). In addition, for moderate individuals there is an age- and gender-based probability that they are available for sampling that day (Table 2). This additional probability is not applied to core-group members. All individuals that are available for pair formation can be sampled to supplement the total number of partnerships in any of the appropriate combinations between core-group members and/or moderate individuals.

From the subset of the population that participates in the pair formation process on a given day, pairs of individuals are randomly sampled, and tested for three necessary conditions for pair formation:

If those conditions were all met, the probability of partnership formation was determined by the age disparity between the partners. To calculate this conditional probability, we use a folded cumulative normal distribution function (CDF) [9], [43] whose mean and variance depend on the age of the woman. The properties of this function are as follows: when the age disparity (male age - female age) corresponds to the mean of the folded CDF, the probability is 0.5, and any deviation from the mean results in a lower probability. To prevent unnecessary computations in the model by rejecting at least 50% of the partnerships, all probabilities generated by the folded CDF are multiplied by two. The equation that defines the mean age disparity, , is described by(1)

This equation is plotted in Fig. 9A (solid pink line). Similarly, the variance of the age disparity can be described by(2)

The shape and parameters of these equations were initially fitted to the observed age disparity and variance in the Dutch sexual survey data, but as these equations represent the preferred age disparity within partnerships of the simulated population, they needed to be subsequently adjusted (by trial and error) such that the resulting age disparity in the model matched that of the sexual survey data (Fig. 9). Among the adjustments were the addition of a necessary condition based on the age of male partner: for men below the age of 18, a partners' age should not be more than 2 years older than their own.

Based on the UK survey data on the duration of the second-most recent partnership (Fig. 5), partnerships in the model are categorized into three types (short, medium and long-term partnerships). Each type of partnership represents an exponential distribution with a minimum duration of 1 day, and a mean duration of days (short), 250 days (medium), and 5880 days (long) (Table 2). The preferred duration of a partnership is determined by first picking the type of partnership from a ratio of 14 (short) to 13 (medium) to 12 (long), and subsequently sampling the exponential distribution associated with that type. As detailed in the previous section, a partnership will only be formed between two individuals if both select the same type of partnership.

The maximum number of partnerships that an individual can simultaneously maintain is described in the model by their sexual capacity, meaning that an individual with a sexual capacity of can maintain simultaneous partnerships. The sexual capacity of an individual reflects how much time, attention, money,etc he/she is willing to invest in partnerships [44].

In the model, the development of the sexual capacity of an individual during his/her life is stored in a vector that is constructed at its birth, by sampling from gamma distributions that determine the age at which an individual starts participating in the pair formation process (and thus when its capacity goes from 0 to 1), and if applicable, the onset, and duration of a period during which its capacity is larger than 1 (Table 1). Individuals with a capacity larger than 1 are part of the so-called “core-group” [23], [24] and are able to start and maintain multiple concurrent partnerships. The maximum capacity of a core-group member is a function of the onset of their core-group period, and the length of that period (Table 1). Core-group members that have a capacity of 5 or higher (about 15% of the core-group), will keep a higher base capacity of 2 after their core-group behaviour period.

Core-group members are not allowed to have more than two non-transitional (i.e. costly, see next section) long-term partnerships at the same time. This constraint makes it possible for some core-group individuals to accumulate up to 600 partnerships over their lifetime, as is observed in the empirical data (see results), and not become tied up in 5 long-term partnerships. When the capacity of an individual drops at the end of its core-group period, he/she will randomly break up a number of partnerships, until he/she is no longer over capacity.

Transitional concurrency (i.e. the period prior to the end of an existing partnership during which an individual acquires a new partnership [25]) is implemented as follows: partnerships that are near the predetermined end date of that partnership no longer carry a maintenance cost, and thus free up capacity for an individual. Transitional concurrency becomes a possibility during the last 15% of the duration of an existing partnership for men, and the last 10% for women. In effect, it allows individuals that have a maximum capacity of 1 to temporarily maintain two concurrent partnerships.

The model has a burn-in period of 60 years, during which a stable sexual contact network is built up in the model population. Ten years prior to the end of the burn-in period, Ct is introduced in the simulated population by infecting 100 core-group individuals with asymptomatic Ct. After 35 years, the average Ct prevalence distribution is measured over a period of 15 years.

The model is implemented in Clojure 1.21, a modern dialect of lisp (http://clojure.org). The source code of the model and an example of the sexual networks that are generated by the model are included as Protocol S1, and Dataset S1 & S2.