Data

Full methodological details of Britain’s third National Survey of Sexual Attitudes and Lifestyles (Natsal-3) have previously been published, and are available from the study’s website: www.natsal.ac.uk, including the Natsal-3 questionnaire [35, 36]. Briefly, Natsal-3 was a complex multi-staged survey involving a stratified, clustered probability sample design. Altogether, 15,162 interviews with people aged 16-74 years resident in Britain were completed between September 2010 and August 2012. The response rate was 57.7%. Computer-assisted personal interviewing (CAPI) was used, with computer-assisted self-interview (CASI) for the more sensitive questions, including those about the number and gender of sexual partners, and whether condoms were used with the partner, and a detailed question module on participants’ three most recent partners. An anonymised dataset is available to academic researchers from the UK Data Service, https://discover.ukdataservice.ac.uk/; SN: 7799; persistent identifier: 10.5255/UKDA-SN-779-2.

Defining partnerships

We first consider the annual number of new sexual partners (where there was unprotected sex, defined from the data as condomless sex on at least one occasion), for individuals who are already sexually active. We begin by assuming that the observations can be generated by each individual having a particular stochastic rate, chosen from a distribution, of acquiring a new partner—hence the partnership dynamics are Markovian and thus can be readily simulated. Note that the number of new partnerships, and not partnership duration, is modelled here—this is discussed further in the discussion.

For a given stochastic rate, the expected number of new sexual partners in a year is therefore assumed to be Poisson distributed. Hence, if the observed distribution of new partners over one year is O₁(partners|characteristics) we simply seek a set of parameters such that: (1) where is the probability of drawing x from a Poisson distribution with mean m. θ (normalised to sum to one) therefore defines the distribution of stochastic rates for each characterised-class c; with the values of θ determined to minimise the difference between the observed and calculated distributions. Surprisingly, simply using m = 0, …, 7 (where m is the mean of the Poisson distribution) is sufficient to capture the heterogeneity in the number of partnerships. We achieve this fitting to the annual data using a maximum likelihood approach for a fixed value of m, and assessed the best m using AIC.

This approach works well for a single year—each individual could choose their stochastic rate from the distribution appropriate for their characteristics. However, over longer timescales this random approach for each year will not work well, repeatedly picking from the distribution would lead to little variation in expected lifetime numbers of partners (as we would be summing across multiple random samples). An alternative solution is to give each individual (at birth) a percentile, which determines the value selected from the appropriate distribution; those with higher percentiles are continually associated with high rates of acquiring new sexual partners (relatively for their age, sex and other characteristics). We therefore define the function Q(P|c) which translates a percentile risk P into a stochastic rate of new sexual partners, for a given set of characteristics. Unfortunately, this fixed percentile approach leads to too much population heterogeneity, as it cannot account for the potential change in individual behaviour over time (e.g. entering or leaving a long-term stable relationship).

As a compromise between lifetime risk percentages and random annual risk percentages, we define an annual probability q of randomly re-drawing a percentile risk (changing sexual behaviour) which depends on individual characteristics (sex and sexual preference) and is a piece-wise linear function of age. This parameterisation is performed by observing predicted numbers of partners over time scales longer than a year (3 years, 5 years and lifetime) and comparing this to the recorded Natsal-3 data. For example, given data on partnerships over 3 years (which is long enough to illustrate the principles, but not so long that there are too many combinations) we predict the following distribution: (2)

For simplicity of notation we have assumed that the same characteristics, c, hold for each individual over the entire 3-year period, although in practice they will age. The three terms in Eq 3 relate respectively to (i) maintaining the same percentile for all three years, (ii) changing percentile once, and (iii) changing percentile twice so that all three years use different values. This allows individuals to change partnership formation rates rapidly, as is commonly observed, at least at younger ages [24].

We infer the parameters that underpin the partnership formation rates (θ) using a maximum likelihood approach, fitting to the annual number of new partners (and new partners without the protection of a condom) in each age-group as given by the Natsal-3 data. Determining the probability of transitions between risk-percentages (q) is more complex, Eq (3) shows the number of terms required when predicting the distribution of partnerships over just 3 years, this obviously increases when one considered the number of lifetime partnerships—especially when the age of first partnership also needs to be taken into account. This restricts our ability to use a likelihood based approach for q; instead we utilise a simulation-based approach generating a synthetic population that can be compared to the data. We take the function q to be a continuous function of age given by a broken-stick model (with 4 free parameters) and use a different set of parameters for each gender. The parameters within the function q are inferred through a ABC-type approach [37], minimising the difference between the observed and simulated distributions of partners over different durations (3 years, 5 years and lifetime).

Natsal-3 recruited participants aged 16-74, although for the spread of STIs the bulk of relevant sexual activity occurs by the age of 50 [24, 38] and we use this age as an upper bound in the model. To account for sexual activity below the age of 16, we need to modify our fitting procedure. The stochastic new partnership formation rates for individuals between 10 and 15 are determined using both the information on age of first sex together with the number of partners over longer times (3 years, 5 years and lifetime), with the expected rates increasing non-linearly up to age 16.

For individuals who have engaged (or will engage) in same-sex activity (about 7% of men report ever having had same-sex sexual experience by the age of 45 [24] the formulation is slightly more complex, as both the number of opposite-sex and same-sex partners need to be defined for each percentile. We do this by a similar method to that outlined above, but partition this population into those who only engage in same-sex activity and those who have sex with both men and women. Thus, for each individual we choose the rate of new partnerships by whether or not the individual has engaged or will engage in same-sex activity. If not, partnership rates are shown in the Results section. For those who engage in same-sex activity, the rates and sex of the partner is determined using distributions in the Supplementary Material.

The above detailed model fitting, only determines one part of the network structure—the degree distribution over time. The other component is to determine who has sex with whom. In principle, this is a large matrix that gives the probability that someone with characteristics and percentile has sex with an individual with characteristics c and percentile P. The characteristics of a partner (in terms of age and sex) are recorded within Natsal-3, in addition we choose the percentile of the partner using a configuration model [39], such that the probability that a partnership is made with someone of percentile P is given by: Q(P|c)/∑_p Q(p|c), assuming the characteristics have already been identified. While this is a useful default assumption, and is consistent with the lack of information on the past behaviour of sexual partners, it could be argued that there will be some level of assortativity between partnerships, with low-percentile individuals (looking for long stable relationships) more likely to partner other low-percentile individuals. However, this element of structure cannot be inferred from the available data (as it would require detailed information of the number of partners of each partner); instead the level of assortativity can only be estimated either by matching to the consequences of the partnership network (in terms of the level and distribution of infection) or from having much more detailed information on the entire partnership network (as seen in small detailed studies [33]).

Defining transmission

The above formulation has determined the number of new (unprotected) sexual partnerships an individual engages with over different periods of time. To translate this to the population scale requires considerable care. Two approaches are feasible. The most intuitive and mechanistic approach would be to explicitly model the full transmission dynamics within partnerships. However, this is fraught with difficulties, most notably that using partnership formation rates calculated above would lead to double-counting of partnerships—as each individual will both generate their own partnerships and be involved in partnerships initiated by others. Hence, to correctly utilise this approach, partnership formation and dissolution at the population level would necessitate a (non-trivial) rescaling of all the parameters defined above. However, if we are prepared to insist that all partnerships occur serially, with no overlapping (concurrent) partnerships, then a simplification becomes possible. Instead of modelling bi-directional transmission within a partnership, we can take an individual-centric approach and only model transmission to an individual every time they form a new partnership; with the rates of partnership formation as specified above. (An alternative way of conceptualising the individual-centric approach is that we model a focal group who pick partners from another (larger) population whose infectious status mirrors that found in the focal group. This ensures that the pattern of partnerships in the focal group matches the Natsal-3 statistics, and that infection is modelled correctly.) These two approaches of rescaled partnership formation rates with bi-directional transmission and an individual-centric methodology with one-directional transmission are identical if concurrent partnerships can be ignored.

We note that our mathematical model makes a number of simplifying assumptions, which greatly reduce the complexity (and dimensionality) of the model and allow us to exploit the heterogeneities captured by the Natsal surveys. The main assumptions are:

• Homogeneous partnerships. We have assumed that all partnerships are associated with equal risk of transmission, however there is considerable heterogeneity between partnerships both in terms of frequency and type of sexual act, which will influence the transmission probability.
• Lack of concurrency. Although our model does not explicitly model concurrent partnerships, it does include such concurrent relationships in the calculation of partnership rates; therefore, the model only fails to capture transmission within the stable partnership due to infection that comes from a concurrent relationship. We believe that inference of the transmission parameter necessary to capture a given level of infection in the population will counteract much of this bias.
• Instantaneous transmission assumption. We have assumed that transmission of infection happens as soon as a partnership begins; this is clearly a simplification but will have a minimal effect on the dynamics as further transmission will not occur until a new partnership has begun. It may however introduce a slight bias in the age dependent level of infection, as individuals will be infected slightly earlier.
• Perfect protection. For simplicity we have assumed that both condoms and vaccination provide complete protection against infection. Both of these simplifications can be overcome by a scaling of either the partnership data (including some partnerships that only have condom protected sex) or the vaccination rate (considering vaccine failure and waning).
• Lack of assortativity. While assortativity appears to be the norm in most social interactions (those with more contacts generally meet others with more contacts), given that the Natsal surveys are individual focused there is no method of obtaining such information from these studies.
• Lack of explicit partnerships. While the lack of explicit partnerships allows our model to be parsimonious, it also provides some restrictions on its use. For example, interventions (such as contact tracing or couple-targeted controls) which are targeted at partnerships are not easily incorporated within our framework. Moreover, information from relationship prevalence data, which is often used to establish transmission probabilities across partnerships, cannot readily be used within our framework.

However, despite these assumptions, we feel that the limitations are more than compensated for by the parsimonious nature of our model.

Stochastic simulations

We simulate the partnership dynamics and transmission of a generic STI using an individual-based stochastic simulation, taking the stochastic rates and probabilities as given above. A fixed population size of 20,000 individuals is used in simulations, which allows rapid simulation without stochastic fluctuations dominating behaviour; this population size is suitable for the parameter ranges used here, although in practice may need to be increased to simulate STIs with very short infectious periods. There are approximately equal numbers in each age class, which is representative of the British population in the age range considered [40]. Individuals are sampled from the population distribution in Natsal-3, and are aged 10-50 (note that Natsal-3 sampled those from age 16, and we infer distributions for those aged 10-15 using the methods above); and as they age through time during simulations, an individual reaching 51 is immediately replaced by one aged 10—thus keeping the population size constant.

Upon an individual forming a partnership, if the partner is infected, then there is a probability of the individual becoming infected instantaneously. We choose a generic STI rather than focusing on a particular STI such as chlamydia, or a particular type such as bacterial or viral STIs. There are reason for this: (i) to allow for basic conclusions to be drawn from this simple example, and (ii) to allow the model framework to easily be adapted for simulating specific STIs. We assume SIS-P (Susceptible—Infected—Susceptible—Protected) dynamics, which is typical for several STIs (particularly bacterial STIs), meaning that after the natural recovery from an infection (or following treatment) that individual is immediately able to be reinfected. Individuals are also able to be protected from birth, meaning they cannot be infected at any time. Again, considering the time-scales involved with many STIs, this is a reasonable and parsimonious assumption as any recovery period is relatively short. We keep the model simple by assuming a fixed probability of transmission when an infected individual partners with a susceptible individual (β, default value 0.7), a fixed recovery rate, the inverse of the infectious period (γ, default value 1 per years), and a fixed rate of protection of individuals ν as they enter the model, which for simplicity we assume offers lifetime protection at 100% efficacy. We vary β, γ and ν in simulations, to observe the effect of disease prevalence both at the population level and within age groups.

Simulations run for 100 years to allow the system to attain the endemic equilibrium. Throughout we use a discrete time step of Δt = 20 days (hence a stochastic rate R is replaced by a probability 1 − exp(−Rδt) that an event occurs within the time step); sensitivity analysis has shown that results are insensitive to a time step of this duration.

Partnership characteristics

For brevity, we here present the results for men and women with only opposite-sex partners; corresponding plots for same-sex individuals are given in the supplementary material, along with the rate by age of opposite-sex partnership formation. These latter distributions are important as STI rates are often higher in groups such as men who have sex with men (MSM) [41], and thus these groups form an important source of infection to the population which models need to capture. Fig 1 shows the parameters that are obtained by matching to Natsal-3 responses. Graphs 1a and 1b show the distributions of the stochastic rate of new sexual partners, Q(P|c), for each percentile risk P and for c being heterosexual males and females. Fig 1c shows the probability that an individual changes their risk percentile (i.e rate of new partnerships) in a given year—low values (at around 28 years old) correspond to consistent individual behaviour (i.e. people tend to remain within their risk group), which in turn maximises heterogeneity at the population level.

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Fig 1. How partnership formation rates change with age.

Plots (a) and (b) show distributions for men and women respectively. Plot (c) shows how the probability of changing the risk percentile for men and women as both age.

https://doi.org/10.1371/journal.pone.0206501.g001

Using these partnership parameters Fig 2 shows a comparison of Natsal-3 survey data (blue) and model simulation (red), displaying both the annual number of new sexual partners without condoms (columns 1 and 3), and the total lifetime number for seven age groups (columns 2 and 4). As expected due to the simple fitting procedure used, the agreement at the annual scale is extremely high; most age-groups rarely form new partnerships and this trend increases with age. The life-time fits are less accurate which may in part reflect participant recall bias or digit preference, with many respondents reporting partnerships that are multiples of five.

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Fig 2. Comparing Natsal-3 data (blue) and the output from model fitting (red).

Data separated into men (left two columns) and women (right two columns). Shown are the annual number of sexual partners without condoms (columns 1 and 3) and total lifetime number of partners (columns 2 and 4).

https://doi.org/10.1371/journal.pone.0206501.g002

Prevalences from stochastic simulations

Fig 3 displays the effects of varying characteristics of the generic STI on the equilibrium level of infection; specifically, the probability of transmission within a partnership β, the average infectious period (1/γ) and the proportion of individuals that are protected ν. For comparison, we first outline the expected results from a simple (homogeneous) SIS-P (Susceptible—Infected—Susceptible—Protected) model, which are based on coupled ODEs: (3) where γ is the recovery rate, ν is the proportion of individuals that are protected before becoming sexually active, μ is the rate that individuals enter or leave the sexually active population (birth rate) and is a modified transmission rate which accounts for both partnership formation and transmission within the partnership. We assume that γ, ν and μ take the same values in both the stochastic simulations and the ODE model, but modify to obtain the same endemic prevalence of infection at the default parameters. This leads to ; we therefore use the relationship (which gives equal prevalence at the default parameters) to compare the two models (i.e. I* from the ODE model (3) to the equilibrium prevalence in the stochastic simulations) as the parameters are varied (Fig 3). We find that the realistic heterogeneities within the simulation model makes the endemic prevalence of infection far less sensitive to the parameter values compared to the simple ODE model.

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Fig 3. How prevalence of a generic STI varies, both at the population level and by age, at the end of simulations, when varying model parameters.

Plots (a), (c) and (e) show the effect of varying transmission probability β, recovery rate γ, and protection rate ν respectively. Dashed lines show the equivalent prevalences using the ODE model given by (3). Confidence intervals are shown by bars. Plots (b), (d) and (f) show prevalence broken down by age at different values of β, γ and ν, with the colour of lines corresponding to particular parameter values in plots (a), (c) and (e). For example, the red line in plot (b) shows prevalence by age when β = 0.3 in plot (a), the yellow line corresponds to β = 0.4, and so on. (Default values are β = 0.7, γ = 1 per year and ν = 0; these are linked to the deterministic model by assuming ).

https://doi.org/10.1371/journal.pone.0206501.g003

We note (although do not show results here) that at the default parameters the ODE model has a much slower growth-rate that the stochastic simulation, despite both having the same equilibrium prevalence; we attribute this to the considerable heterogeneity with the simulations such that a small high-risk group can rapidly transmit infection when it initially invades. In addition, using a more complex ODE model that is homogeneous in terms of risk, but recognises partnership formation [42] does not change the qualitative results of this comparison.

In addition, Fig 3 shows the age-distribution of infection from the simulation model. As might be expected the highest prevalences of infection are in individuals aged 16-25, which is a direct consequence of the Natsal-3 data, as partnership formation rates tend to be highest in this age group [24].