Article Text

## Abstract

**Background** Models of sexually transmitted infection (STI) transmission can offer insights as to why gonorrhoea and other STIs are disproportionately concentrated in epidemiologically distinct subpopulations.

**Methods** We highlight two different constructs for modelling STIs by drawing on previously published work on pair and metapopulation models, and reanalysed partnership data from the National Survey of Sexual Attitudes and Lifestyles II (NATSAL II) in the UK.

**Results** Pair models account for intrapair reinfections and are necessary to illustrate the importance of partnership dynamics. The pair modelling framework suggests that a key determinant of transmission is the length of time or ‘gap’ between partnerships, and that partnerships of medium length can potentially be more efficient for gonococcal transmission than the shortest partnerships. As for the metapopulation framework, one key insight is that the epidemiology of gonorrhoea is possibly being driven by subpopulations with higher than average concentrations of individuals with high sexual risk activity. The reanalysis of data on sexual behaviour in the UK shows that well recognised population subgroups at higher risk of gonorrhoea do also have higher levels of risk behaviour, such as a higher average number of new partners per year, as well as a higher prevalence of concurrent partnerships and short gaps before partnerships.

**Results** The concentration of risk behaviour in key population subgroups may be leading to self-sustaining pockets of transmission for STIs. Combinations of partnership behaviours at the level of population subgroups should be a subject of future empirical research as well as modelling efforts.

- Sexual networks
- mathematical model
- gonorrhoea
- epidemiology
- sexual behaviour
*Neisseria gonorrhoea*

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## Introduction

Studies of the epidemiology of gonorrhoea and other sexually transmitted infections (STIs) demonstrate that some subpopulations are at much higher risk of infection than others. For example, in the UK1–6 and elsewhere,7–11 gonorrhoea is disproportionately concentrated in sociogeographically definable groups, such as young people and some ethnic minorities, with spatial clustering observed within urban conurbations. Higher concentrations of at-risk ethnic minority groups in these locales explain some but not all of the observed geographical clustering.5 12 Different degrees of sociogeographic clustering occur with different STIs,4 8 11 suggesting that explanations for STI epidemiology may lie in aggregated behavioural and partnership characteristics at the subpopulation level, so that certain STIs can find an ecological niche in some subpopulations but not in others.

Models of STI transmission can help us to develop our understanding of these observations, and thus provide a better foundation on which to model the effect of potential interventions.13 The original model of STI transmission proposed by Hethcote and Yorke14 suggested that gonococcal transmission could be explained by stratifying the population into sexual activity classes, with a ‘core group’ having a higher rate of change of sexual partners than the rest of the modelled population. However, these simple ‘activity class’ models provide a poor description of gonococcal transmission in the low prevalence settings of industrialised countries.15 In particular, if the disease was indeed equally distributed at such a low prevalence as predicted by activity class models, the infection would be near the threshold where it can persist, so that elimination of the infection should be easy.

We performed two studies16 17 to better understand the limitations of the activity class models and identify additional aspects of the sexual network structure behind patterns of STI transmission. Firstly, pair models, which can account for reinfection within partnerships,18 19 were used to examine the respective role of partnership lengths and the time between partnerships (the ‘gap length’) on gonococcal transmission.16 Secondly, we demonstrated that a ‘metapopulation’, comprising multiple interconnected subpopulations, could be used to summarize sociodemographic and spatial stratification, resulting in much more realistic model outputs than the original activity class model.17

In this article, we revisit our pair modelling work and elaborate on the importance of considering within partnership transmission. We also review our metapopulation work and reanalyse data from the National Survey of Sexual Attitudes and Lifestyles II (NATSAL II)20 to show how partnership behaviours can possibly account for the metapopulation effect in gonococcal transmission. Finally, we discuss how to further extend the metapopulation concept to partnership-based or individual-based models to understand the full role of the sexual network in determining patterns of STI transmission.

## The importance of intrapair dynamics

Pair models for STI transmission were proposed by Dietz and Hadler more than 20 years ago.19 Such models divide the population into pairs and singles. Pairs form when two singles pair up, and pairs dissolve to give two singles. Transmission occurs only in pairs of individuals with discordant infectious states. In the simplest case of a model with monogamous pairs and a fixed gap and partnership behaviour, we can define the cycle length, *L ^{C}*, as the sum of the gap (

*L*) and partnership (

^{G}*L*) lengths. The partner change rate (

^{P}*c*) for such a model is then the inverse of the cycle length (ie,

*1*/

*L*). A mathematical description of the model is given in Appendix 1.

^{C}Pair models are, intuitively, a more accurate depiction of the process behind STI transmission. While the classical model assumes that partnerships are instantaneous, pair models account for time spent within partnerships and time spent between partnerships (the ‘gap’). Lloyd-Smith and colleagues previously showed that, except in the extreme case where cycle lengths are very short, the classical model produces higher prevalence than the pair model for STIs with short infectious periods.21 Kretzschmar, in her review of pair and network models,22 gives a qualitative explanation for this. Firstly, she says, ‘susceptible persons who are in a long-lasting partnership with another susceptible individual are effectively protected from infection’, and secondly, that ‘infected individuals paired with another infected individual waste contacts without infecting others’. In addition, pair models account for reinfection within partnerships.18 Figure 1A illustrates why this is important. Suppose an uninfected woman pairs with an infected man and subsequently acquires the infection. The man recovers, but is then reinfected and consequently remains infectious and able to infect the next partner when the pair separates. In contrast, the contribution of repeated infection is unaccounted for in the activity class model. This can be inferred from the classical expression for the reproductive number, *R _{0}*=

*bcd*, where

*c*represents only contact with new partners (

*b*being per partnership transmission probability and

*d*the infectious duration), and ignores intrapartnership contacts and reinfections. Within partnership reinfection substantially extends the effective infectious period of an infectious case while within a pair. For instance, figure 1B simulates a hypothetical woman starting in the symptomatic and care-seeking state using gonorrhoea-like parameters. If this individual were single, or if recovery were simulated using the classical model, she would have an almost 100% probability of recovery within 3 months. However, if she were in a partnership, she would have a 40% chance of remaining infectious after 3 months. Not surprisingly, per sex act transmission probabilities affect the chance of remaining infectious, since more efficient transmission (higher

*b*) increases reinfection rates. However, even with perfect transmission within pairs, because there is always a gap between consecutive episodes of sex (3 days in figure 1B), the probability of remaining infectious will decrease with increasing partnership length, since both members of the monogamous pair will ultimately recover.

These pair dynamics have important implications for the persistence of infection in the population. Firstly, if the gap length is a fixed proportion of overall cycle length (figure 2A), as expected, we see a near linear decrease in prevalence with longer partnership lengths since an increasing proportion of infected pairs recover without passing the infection onwards. However, compared to a higher gap/cycle ratio of 0.8, a lower gap/cycle ratio of 0.2 gives a much higher prevalence and persistence over a wider range of partnership lengths, because more time spent in partnerships prolongs the effective duration of infectiousness. Secondly, an intuitively unusual finding is observed when the gap length is fixed (here at 10, 30 and 50 days). Since the gap as a proportion of the overall cycle length now decreases as partnership lengths increase, we instead see that prevalence is highest at an intermediate partnership length (figure 2B).

The result of such intrapair dynamics makes mid-length partnerships (*L ^{P}* of several months) reasonably efficient for transmission. These partnership dynamics can also give rise to persistence of the infection at low partner change rates, since

*c*=

*1*/

*L*=

^{C}*1*/(

*L*+

^{G}*L*). For example, in figure 2C, we plot the longest gap length, which supports transmission in a monogamous pair model across a range of partner change rates. With gap lengths of less than 1 week and low levels of condom use (10%), we see that very low partner change rates of about 0.6 per year can possibly sustain infection for gonorrhoea-like parameters. However, at high partner change rates, short gap lengths are still needed as the gap is now limited by the cycle length itself. For instance, when

^{P}*c*=40 per year (eg, in Garnett

*et al*),15 even if the partnership were instantaneous, then the maximum gap length able to sustain transmission would still only be 365 days/40, or about 9 days. Therefore, when framed in the context of partnership dynamics, a critical quantity of interest in driving transmission is the gap rather than the partner change rate, since individuals with low partner change rates but short gaps can support transmission. In addition, the existence of individuals with short gaps but relatively low partner change rates might explain why activity class models have overestimated and underestimated relative gonococcal incidence in high and low activity individuals, respectively.15

Using these concepts coupled with data on reported partnership lengths and gaps from NATSAL II, we previously showed that these dynamics have important implications for interventions.16 For example, decreasing the proportion of partnerships that last a single day has little impact on incidence. This is because gap lengths in such partnerships are long relative to the short partnership length (high gap/cycle ratios), making such behaviour less conducive to transmission. Instead, the most effective strategy for gonorrhoea control would be to focus on reducing the proportion of individuals with short gaps. We also found that condom use must be increased in short and mid-length type partnerships to obtain substantial reductions in incidence rates.

However, there are several challenges to be overcome to fully understand the importance of partnership dynamics on STI transmission. Firstly, other than NATSAL II, we know only of a few other studies23 24 with detailed partnership data. In particular, it is the correlation between partnership length and gaps that need to be better understood in order to more appropriately target interventions. Secondly, as has been recently pointed out by the investigators who designed NATSAL II, analysis of such partnership data needs to appropriately weight current and past partnerships, since current and longer partnerships have a higher chance of being reported in detail.25 26 Moreover, the pair model we proposed could not account for the differential behaviour of men and women, and the effect of concurrent partnerships. In the context of concurrent partnerships, gaps could be seen as the time between sex acts with two consecutive existing partners; the average time between sexual acts with two existing partners was as short as 8 days in one at-risk population.27 In addition, concurrency would also have other effects, such as further augmenting the infectious duration through reinfection within triplets, and by allowing transmission into and out of longer partnerships. The pair modelling framework can be extended to incorporate concurrent partnerships,28 but modelling triplets and higher order configurations of concurrency in addition to multiple gap and partnership categories would vastly increase model complexity. Finally, as for all models of STI transmission, the results from pair models are highly sensitive to assumptions about sexual mixing and changes in behaviour over the life course of individuals. This suggests that additional social structure may also be important in explaining the transmission dynamics of STIs.

## The rationale for a metapopulation framework

Given some of the limitations of pair models in fully describing gonococcal transmission, we proposed a metapopulation framework for modelling STIs.17 The concept of a ‘metapopulation’ is borrowed from the field of ecology, and implies a ‘population of local populations’;29 that is, a population composed of several subpopulations that are linked in some way to each other. These might describe geographically distinct areas, or alternatively social groupings within which sexual encounters occur which may or may not overlap geographically. In our metapopulation model, the population is divided into subpopulations where sexual partnerships are more likely to form between members of the same subpopulation than between members from different subpopulations. The key distinction between the subpopulations lies in their composition. For instance, in figure 3A, subpopulation 2 has more high activity individuals than subpopulation 1.

Since the prevalence of gonorrhoea in the general population is low, in a metapopulation model, the chance of contact with an infectious case of gonorrhoea depends largely on the subpopulation membership, and contacts outside the subpopulation are essentially wasted. The existence of subpopulations with high concentrations of high activity individuals thus generates conditions conducive to ‘local’ persistence. Within a subpopulation, in the absence of reintroduction, the transmission dynamics are essentially similar to those predicted by the original activity class models. Thus to sustain transmission, a subpopulation will need a high proportion of high activity individuals, which results in a high prevalence of infection within the subpopulation. However, since these subpopulations form only a small proportion of the total population, the overall prevalence remains low. Moreover, high activity individuals in subpopulations without self-sustaining transmission are ‘protected’ by being isolated from infectious individuals, whereas the original activity class model treats all high activity individuals equally.

These differences give rise to more realistic dynamics than the original activity class model. When both models are parameterised to give the same incidence of infection and then seeded with the infection, it took more than 10 times as long to reach the equilibrium incidence in the activity class model as compared to the metapopulation model.17 Moreover, small parameter changes such as the time to treatment for symptomatic cases translate into a precipitous decrease in the equilibrium incidence in the activity class model but a more modest near-linear decrease in the metapopulation model (figure 3B). The metapopulation framework also provides a plausible explanation for several key features of gonococcal and STI epidemiology. For instance, incidence in certain subpopulations can be many times the population average in a manner which mirrors the extreme incidence rates observed in key at-risk populations.1 3 30 Figure 3C also shows how the metapopulation framework provides a logical explanation for the skewed geographic distribution of cases for gonorrhoea as well as other STIs,4 8 11 with STIs having lower overall transmission potential being more skewed in their distribution and vice versa.

If indeed the metapopulation conceptualisation is correct, then it might be possible to observe correlations, at the level of subpopulations, of risk factors for gonorrhoea and other STIs. Such risk factors, as identified through the activity class and pair models would be higher partner change rates and short gaps and concurrent partnerships, respectively. We reanalysed the NATSAL data to identify the proportion of individuals who reported either concurrency or a gap length of 2 months or less16 in their most recent partnership. Figure 4A shows that well recognised risk factors for gonorrhoea such as residence in Inner London, younger age and black ethnicity1 were indeed associated with higher levels of concurrency as well as short gaps and mean number of new partners in the past year. However, they also had higher levels of condom use. Figure 4B stratifies the data by age and ethnicity. We see that close to 30% of individuals aged 15 to 24 of black ethnicity had concurrent partnerships or short gaps associated with their most recent relationship, as well as higher partner change rates. The higher partner change rates in these population subgroups therefore suggest why age-stratified and ethnicity-stratified models have worked better for gonorrhoea than models stratified by activity class alone.31 32

## Discussion

The pair and metapopulation frameworks both provide insight into which aspects of the social and sexual network are important determinants of STI transmission. Firstly, pair models highlight the importance of reinfection within partnerships in lengthening the infectious period. This phenomenon may be overestimated in the pair model, since contact tracing or expedited partner treatment should increase the chance of infected pairs recovering simultaneously. However, the proven efficacy of expedited partner treatment33 is indirect evidence that reinfection is an important phenomenon in gonorrhoea and chlamydia epidemiology. Secondly, pair models demonstrate that an additional factor, namely the length of the gap between the partnership dissolving and new partnerships forming, is critical to transmission. Concurrency can also be thought of in this manner as the ‘gap’ between sexual acts although understanding the complex pathways associated with transmission in concurrent partnerships requires further theoretical exploration. Drawing insight from our metapopulation framework, it is of interest to understand how partnership characteristics in subpopulations defined on the basis of demographic characteristics typically associated with STI risk may differ. Using UK behavioural data from NATSAL II, our initial analyses show that the proportion with short gaps before partnerships, the proportion with concurrent partnerships and the mean number of partners in the past year was higher in population subgroups known to be at higher risk of gonorrhoea and other STIs. Short gaps have also been found to be associated with age and ethnic strata and history of previous STIs elsewhere.34 Since incidence rates of gonorrhoea in young people from black minority ethnic groups can be about 20 times or more that for the general population,2 it is unlikely that the higher prevalence in these population subgroups is simply due to the sum of increased individual-level risk behaviour. Rather, in line with our metapopulation hypothesis, the high proportion with concurrent partnerships and short gaps in individuals aged 15 to 24 of black ethnicity (figure 4B) likely facilitates the formation of pockets of self-sustaining transmission, giving rise to an exponential (rather than linear) increase in risk. Further empirical research on the correlations between gaps, partnership lengths and STI risk in different populations is needed in other populations.35

A metapopulation conceptualisation of STI dynamics has obvious implications for targeting and focusing interventions for different STIs. It also has theoretical ramifications for modelling. Ethnicity, age and area of residence have been found to be associated with genetic clustering of gonococcal strains.5 36 These results provide further support for our metapopulation hypothesis as they suggest circulation of strains within subpopulations. Taken together, the empirical findings and theoretical insights suggest that the high prevalence of short gaps, concurrency and appropriate partnership lengths in key sociogeographic groups are resulting in metapopulation-like effects through the formation of sexual networks with emergent properties,37 namely, the ability to function as host populations for transmission of different STIs. Recently published work using network simulation models indeed suggests that concurrent partnerships may be acting together with assortative mixing to drive the high prevalence of HIV and other STIs among African–Americans,38 providing further support for the concept of high risk subpopulations as proposed by the metapopulation framework. This work represents a logical advancement over previous efforts to build age and ethnic stratified ‘activity class’ type models, given the inability of ‘activity class’ models to adequately account for the dynamics of intrapair reinfections, which we argue to be of importance in susceptible–infectious–susceptible type pathogens. Additional modelling work that combines partnership characteristics with sociogeographic factors is needed to better understand the importance of overlap between populations and risk behaviours. These however must be closely linked to further empirical data collection on gap and partnership behaviours at the level of different subpopulations, so that future models can better identify and prioritise interventions for future STI control.

### Key messages

Pair models suggest that, with short gap lengths, intrapair reinfections can sustain transmission of susceptible–infectious–susceptible infections such as gonorrhoea at low partner change rates

Metapopulation models of sexually transmitted infections (STIs) suggests that concentrations of risk behaviour at the subpopulation level drives commonly observed disparities in STI incidence

Empirical data shows that risk behaviours such as short gaps, concurrent partnerships and higher partner change rates are more prevalent in high STI incidence subpopulations

STI models incorporating appropriately weighted empirical data on gaps and concurrent partnerships at the subpopulation level should be considered, possibly using individual-based modelling approaches

## Acknowledgments

The corresponding author would like to thank the Department of Epidemiology and Public Health, and Associate Professor Wong Mee Lian at the National University of Singapore for hosting him at the time when some of the work for this paper was completed.

## Appendix

In figures 1 and 2, gonorrhoea was modelled as susceptible–infectious–susceptible (SIS) type infections, with two infectious states, representing individuals that seek care from symptoms (state *Y*, shorter infectious period), and individuals that do not seek care from symptoms (state *Z* longer infectious period). A heterosexual population of size *N* with an equal number of men and women is modelled. *ω* and *ϕ* are pair separation and pair formation rates equal to the inverse of the partnership (*L ^{P}*) and gap (

*L*) lengths, respectively. Individuals recover without mortality to become uninfected (state

^{G}*X*).

*M*,

_{X}*M*and

_{Y}*M*are men, and

_{Z}*F*,

_{X}*F*and

_{y}*F*are women, where the lettered subscripts denote the respective infection status. In pairs, the first and second subscripts denote the infection states of the man and woman, respectively, (eg,

_{Z}*P*,

_{X}_{X}represents pairs where the man and woman are uninfected). Lettered superscripts are used to distinguish gender specific parameters, to allow for gender differences in per sex act transmissibility (

*β*,

^{M}*β*), the rates of recovery from infection

^{F}*α*and

^{M}*α*are the per unit time transmission probabilities, where:

^{F}where *ζ* is the average daily frequency of sex within a partnership, so that *α*=*ηβ* when sex occurs once a day (the maximum frequency allowed in the model). *η* is a factor adjusting for reductions in per sex act transmission probability due to condom use (*χ*), which is assumed to be 70% efficacious, that is:

The equations describing the model are:

The prevalence of infection in the pair model would be:

Figure 3 was based on models and parameters described in detail in our previous work.17 The parameters assumed for gonorrhoea also follows those assumed in our previous work16 17 and are given in the table AI below.

Parameter values | ||||

Parameter | Symbol | Men | Women | |

Frequency of sex | ς | Once in 3 days (1/3) | ||

Duration of infectiousness: | ||||

Seek care from symptoms | 13 days | 20 days | ||

Do not seek care from symptoms | 185 days | 185 days | ||

Proportion of new infections: | ||||

Seek care from symptoms | 0.59 | 0.36 | ||

Do not seek care from symptoms | 0.41 | 0.64 | ||

Transmission probability: | Male to female | Female to male | ||

Per sex act | β^{M}, β^{F} | 0.50 | 0.25 | |

Per partnership | γ^{M}, γ^{F} | 0.80 | 0.60 |

## References

## Footnotes

Funding MIC was partially funded by the ExxonMobil-NUS Research Fellowship.

Competing interests None.

Provenance and peer review Not commissioned; externally peer reviewed.