# Mind the Gap: The Role of Time Between Sex With Two Consecutive Partners on the Transmission Dynamics of Gonorrhea

Objective: Both the duration of sexual partnerships and the time between two consecutive partnerships (gap length) varies between populations. We use a mathematical model with multiple partnership durations and gap lengths to identify the types of relationship cycles that sustain gonorrhea transmission in the United Kingdom.

Study Design: A mathematical model for gonorrhea transmission was constructed which tracks the duration of partnerships and their preceding gap lengths. The National Survey of Sexual Attitudes and Lifestyles was used to parameterize the model population into 5 different partnership lengths (mean of 1 day, 2 weeks, 8 weeks, 30 weeks, and 10 years) and 3 preceding gap lengths (14 days, 8 weeks, and 1.5 years).

Results: The model was able to reproduce patterns of gonococcal infection in the United Kingdom. Assortative (like-with-like) mixing of individuals with short gaps between partnerships was required for gonorrhea infection to persist. Prevalence was highest in individuals with short (>1 day–<1 month) and midterm partnership durations (>1 month–<3 months). Interventions (such as increased condom use) targeted at those with medium-term partnerships were most effective at reducing prevalence; in contrast targeting interventions at those with short partnerships but longer gap lengths (i.e., the group with the highest number of sexual partners) had relatively less impact.

Conclusion: Our model suggests that gonorrhea is sustained by the presence of a small group of individuals with short gap lengths and medium length partnerships. Interventions targeted at this group are more effective than those targeted at individuals with high numbers of sexual partners but longer gap lengths.

A pair model of sexually transmitted infections shows how gonococcal transmission is driven by individuals with short gaps between partnerships, but not necessarily those with the highest partner change rates.

From the *Centre for Infections, Health Protection Agency, London, UK; †London School of Hygiene and Tropical Medicine, London, UK; and ‡Department of Infectious Disease Epidemiology, Imperial College, London, UK

Correspondence: Mark I. Chen, MPH, Modelling and Economics Unit, Health Protection Agency, Centre for Infections, 61 Colindale Avenue, London NW9 5EQ, UK. E-mail: Mark.Chen@lshtm.ac.uk.

Received for publication January 31, 2007, and accepted November 5, 2007.

ALTHOUGH THE INCIDENCE OF gonorrhea in the United Kingdom has decreased over the past 3 years, it remains high with a total of 19,284 diagnosed cases in 2005.^{1} Underlying this figure are marked differences both geographically and by ethnic group and age. Over 50% of gonorrhea cases are observed in London^{1} with incidence rates highest in young black Caribbean and black African people (estimated annual incidence in 2004: 392 per 100,000) and in men who have sex with men, (estimated annual incidence in 2004: 1834 per 100,000 population).^{2} Control of gonorrhea in the United Kingdom continues to focus on provision of treatment through a network of genitourinary clinics including partner notification and in some areas more active contact tracing. However, it is increasingly being acknowledged that such an approach alone may be inadequate and that an additional risk-based approach that targets populations at high risk of disease may be needed.^{2–4}

The existence of high risk groups whose presence maintained gonorrhea in populations – so-called “core groups” – was first raised in theoretical studies over 30 years ago.^{5} Whilst the hypothesis of such groups is now widely accepted, there has since been significant debate as to how such groups can be identified. Most mathematical models define these groups on the basis of sexual partner change rates, which are then generally equated with numbers of sexual partners.^{6} Thus those with high numbers of sexual partners are assumed to be the group maintaining infection in the population. However, this simple extrapolation remains at odds with empirical observations. For example, although those of black Caribbean ethnicity in the United Kingdom have a 10-fold higher risk of acquiring gonorrhea than those of white ethnicity,^{2} the differences in numbers of sexual partners between these two ethnic groups are small, and have been found to be an insufficient explanation for the difference in the risk of sexually transmitted infections (STIs).^{7}

Here we re-examine the assumptions behind the original models, in particular, the assumption that sexual partnerships are “instantaneous”. As shown in Figure 1a, a single relationship cycle involves ending a partnership, a period spent between partnerships (referred to as the “gap”) followed by the initiation of a new partnership within which sexual activity occurs. Although the partner change rates assumed in the original models are related to the average length of the relationship cycle, relationship cycles of the same length can be composed of different combinations of gap and partnership lengths (Fig. 1a). Longer partnerships favor transmission through increased opportunities for sexual contact, while longer gaps do the opposite by providing more time for recovery from infection between partnerships. Because models based on partner change rates fail to distinguish the relative impact of gap and partnership lengths on STI transmission, they may inaccurately depict the dynamics of certain STIs; in particular, it has been suggested that the effect of partnership lengths are of importance to STIs with a low transmissibility such as human immunodeficiency virus,^{8} whereas the effect of gap lengths are critical to STIs with a short infectious period^{9} such as gonorrhea. A recent study showed that there is a wide variability in both partnership lengths as well as the gap between partnerships.^{10} We therefore constructed a mathematical model for gonorrhea infection which incorporates multiple partnership durations and gap lengths and use this model to identify the groups sustaining gonorrhea transmission on the basis of their relationship cycles.

## Materials and Methods

### Mathematical Model

We modeled a heterosexual population cycling sequentially between being single and being in partnerships. For simplicity, we ignored overlapping partnerships (i.e., concurrency) and modeled only pairs. We also assumed that the number of sexually active men and women were equal, and that the overall population and its distribution into singles and partnerships were constant over time, with no loss or entry of individuals.

We considered 5 partnership and 3 gap lengths. Individuals were assigned to a particular gap-partnership combination and at any point in time could be either single or in a partnership with the rates at which individuals form and leave partnerships defined as the inverse of their gap and partnership lengths, respectively. Within partnerships, the preceding gap lengths of the male and female member are tracked resulting in 9 partnership subcompartments for each partnership length (Fig. 1b). The pairing of males and females of different preceding gap lengths is determined by a mixing parameter ε; ε=1 denotes fully assortative pairing (e.g., males with short gaps pair only with females with short gaps) whereas ε = 0 denotes pairing of males and females proportionate to the availability of opposite gender forming partnerships. Individuals were allowed to change their gap-partnership preference at the end of a partnership; this was determined by ρ, the proportion of individuals leaving a partnership who cross-over to a different gap-partnership combination at the start of the next relationship cycle.

Infected individuals could either be symptomatic and care-seeking or asymptomatic or fail to seek care (noncare-seeking infections). Recovery from either infectious state returns individuals to a susceptible and uninfected state. The infection states of the male and female members of a pair are tracked separately, resulting in 9 possible combinations of infection states for partnerships (Fig. 1b). Recovery occurs in partnerships as well as amongst the singles, whereas infections occur only in partnerships where one member is infected and the other susceptible. The infection rate in such partnerships is the product of the probability of disease acquisition per sex act, the proportion that did not consistently use condoms, and the frequency of sexual intercourse within partnerships. The model was implemented in Berkeley Madonna version 8.3. The mathematical description of the model and the formulae for deriving the population structure are detailed in the appendix.

### Parameters for Population Structure and Sexual Behavior

Data from the National Survey of Sexual Attitudes and Lifestyles (NATSAL 2000), a cross-sectional population-based survey of individuals aged 16 to 44 in the United Kingdom,^{11} was used to obtain parameters for the duration and gaps between partnerships. We excluded individuals reporting any lifetime homosexual partners and factored in weights for the core sample in NATSAL 2000 for all calculations. Dates of first and last sexual intercourse for the 3 most recent partners were used to derive the distributions of gap and partnership lengths without distinguishing between genders (a simplifying assumption made to avoid the difficulties of differential reporting between males and females). Gap lengths were based on the time between the first sexual intercourse with the most recent partner and the last sexual intercourse with the second most recent partner. Only 1.6% of individuals reported these 2 sexual encounters as occurring in the same month; these could either have had a very short period of transitional concurrency,^{12} or a short gap of between 0 and 30 days; we therefore arbitrarily assumed a gap length of 14 days for this group. While those with partnerships which overlapped for a longer period could also be thought of as having short gaps, other indicators of sexual behavior (e.g., number of partners in the past 3 months and partner change rates) suggested that these individuals were different from those with the shortest gaps; these individuals were therefore excluded when deriving our population structure. Exploratory analysis showed that gap lengths of up to 4 months could potentially support transmission, so we defined individuals reporting these gap lengths as a group with midlength gaps. Finally, individuals reporting gap lengths of ≥4 months were considered to have long gaps. In addition, individuals with only 1 life-time partner and those with missing gap lengths were closest in behavior to those reporting long gaps; we therefore included these individuals in the group with the longest gaps, which comprised the majority of the population (91.5%).

The distribution of individuals engaging in partnerships of various lengths could not be directly measured from all sampled individuals, as reported partnership lengths are right-censored in cross-sectional studies (e.g., individual D in Fig. 1). We therefore derived the distribution of partnership lengths from a Kaplan-Meier analysis of the most recent partnerships, with survival time being the duration between the dates of first and last sexual intercourse, and time-zero being the first sexual intercourse. We divided the population into 5 partnership lengths (see Table 1) for direct comparison with a study on heterosexual gonococcal infections in London.^{13} The distributions of gap and partnership lengths derived from the data were then combined to estimate the proportion of the population who would be in each gap-partnership combination at any point in time (Table 1).

### Parameters for Frequency of Sexual Intercourse, Condom Use, and Gonococcal Infection

Partnerships lasting a single day were assumed to comprise 1 episode of sexual intercourse, whereas in longer partnerships sexual intercourse was assumed to occur an average of 2.44 times per week as observed in NATSAL.^{11} Nineteen percent of NATSAL respondents also reported consistent condom use in the prior 4 weeks.^{11} We assumed per sex act transmission probabilities of 0.25 from women-to-men,^{14} and 0.5 from men-to-women.^{15} The durations of infectiousness followed those in Garnett et al.,^{6} being 13 days and 20 days in symptomatic care-seeking men and women, respectively, and 185 days in individuals of either sex who did not seek care. Estimates of the proportion of individuals who seek care were obtained from a study using clinic and community-based case-finding methods^{16} that suggested that 59% of new infections in men and 36% of new infections in women are symptomatic and care-seeking. Observed cases in the clinical setting were these symptomatic care-seeking cases (89% of men and 53% of women seen in the clinical setting^{17}) plus those that are brought to care for other reasons (assumed to come from the asymptomatic/noncare-seeking pool in our model). The parameter values used are summarized in Table 2.

### Comparison of Model Output With Observed Data

Incidence was expressed as rates per 100,000 per year. Our modeled incidence is compared to the reported number of gonococcal infections in England and Wales in 2004 in the 16 to 44 age-group excluding those attributed to men who have sex with men.^{1}

## Results

Figure 2 shows the partnership duration and gap-length combinations in which gonorrhea is able to persist in our model if there is no variation in the behavior of the population (i.e., all individuals have the same partnership and gap lengths). If average partnership lengths are short in the population, then gaps between partnerships must also be short for gonorrhea to persist. Similarly, if partnership lengths are long, then gaps between partnerships must also be short to ensure that infection can continue to be transmitted. In contrast, if partnership lengths have an intermediate duration (between 50 and 150 days) then gonorrhea can persist for a wider range of gap lengths, with a maximum gap length of approximately 65 days allowing persistence if partnership durations are between 50 and 75 days.

Table 1 shows the distribution of partnership and gap lengths estimated for the UK population aged 16 to 44. Approximately 10% of all partnerships are short-term (less than 1 month) and 85% long-term (1 year or more). The majority of new partnerships (91.5%) are estimated to be formed more than 4 months after the preceding partnership ceases. Thus, on the basis of our simple model (Fig. 2), only a small part of the population (those with gap lengths of less than 4 months – 8.5% of the population) could play a role in sustaining gonococcal transmission in the United Kingdom.

Using these values in our model, and assuming that partnership length and gap lengths are independent (a conservative assumption), Figure 3A shows how the prevalence of infection is determined by how those with different gap lengths might form new partnerships. For the infection to persist, some degree of assortative mixing between individuals with short gaps (i.e., those who have recently split from their partners being more likely to form partnerships with others who have recently split) is necessary. However, under the range of mixing assumptions which allow persistence, less assortative mixing means that the infection is able to spread beyond individuals with short gaps to those with midlength and long gaps and thus result in a higher prevalence. For all scenarios, prevalence is highest and is hence sustained by partnerships of short-term and medium duration (1 day–3 months duration). The prevalence in these partnership lengths is similar to estimates from a screening program which targeted at-risk youths within the United Kingdom.^{18} Single-day partnerships and partnerships of more than 3 months could potentially support gonococcal transmission but only under assumptions of moderately assortative mixing. Long-term partnerships could not support transmission regardless of the mixing assumptions although infection in the partnerships can still occur.

Figure 3B shows the distribution of infections by partnership type predicted by the model if we allow cross-over (ρ) between gap-partnership combinations. Increasing the degree of cross-over increases the proportion of infections which are attributed to long-term partnerships and also reduces the overall intensity of transmission. Thus, even with highly assortative mixing by gap length (ε = 0.9 in Fig. 3B), infection was unable to persist if more than 20% of individuals cross-over to a different partnership-gap length combination at the end of each partnership. Although our model was unable to closely replicate the patterns of infection in the female population, we found that with values of ε = 0.9 and ρ = 0.15, it could reasonably reproduce the observed incidence and patterns of infections in the male population (Fig. 3B). Under these mixing assumptions, short-term partnerships (1 day–1 month) are predicted to be the source for about 50% of infections (Fig. 3B) whilst more than 60% of all infections would be caused by pairing with an individual who had a short gap before a partnership (data not shown).

Figure 4A shows the effect of changing the size of groups with different sexual behaviors as might occur from interventions promoting behavioral changes. Reducing the proportion engaging in single day partnerships can have an apparently paradoxical effect of increasing incidence marginally if there is a corresponding increase in the proportion who engage in short and midterm partnerships. Reducing the proportions in short and midterm partnerships results in a relatively small decrease in incidence. The most dramatic reduction in incidence is achieved if the proportion of partnerships formed after short gaps is decreased, with over an 80% reduction in incidence if the proportion of the population with such gaps is reduced by 1%. Figure 4B shows the impact that increasing condom use could have. If such a policy is targeted at those in the short (1 day–1 month) and midterm partnership categories, incidence can be dramatically reduced, although condom use would have to be simultaneously increased in both groups to fully eliminate transmission.

## Discussion

Our results demonstrate that the patterns of partnership duration and gaps between these partnerships are critical in sustaining the transmission of gonorrhea. Partnerships that are of short to medium duration (from >1 day–3 months) appear to play the greatest role and allow gonorrhea to persist for a wide range of gap lengths. When coupled with gap lengths that are short enough to ensure onward transmission plus a high degree of assortative mixing between people in these types of relationship cycles, these partnership patterns are able to reproduce the patterns of prevalence of gonorrhea observed in the United Kingdom.

Perhaps somewhat surprising is the conclusion that the very shortest partnership lengths (those of single day duration) play a less important role in maintaining gonorrhea transmission. This is simply because such partnerships are less likely to be followed by new partnerships sufficiently rapidly to ensure onward transmission. Consequently, individuals with the highest partner change rates may not always be the most critical group in the transmission of an STI, a result that contrasts with the classic definition of a core group. Instead, our results suggest that the “core-group” might be the small number of individuals who have short gaps with the appropriate partnership lengths, estimated here to be approximately 8.5% of the population.

One previous study demonstrated that women who ever had an STI had, on the average, shorter gap lengths than those who never had an STI.^{19}. Also, other studies have shown an increased risk of infection in individuals with 2 or more partners in the past month.^{13,20,21} Such individuals must either be in concurrent partnerships or have a gap of less than a month between consecutive partners and hence could be considered members of our revised core-group. Furthermore, a case-control study showed that individuals infected with gonorrhea had a median partnership length of 3 to 4 months, significantly shorter than that observed in general population controls who had median partnership lengths of approximately 3 years.^{22} Whilst empirical studies^{23–25} suggest that sexual mixing is largely assortative, little is known about mixing between individuals of different preceding gap lengths, or on the proportion of individuals who sequentially cycle through short to midlength partnerships with short gaps. Given our findings, it is useful to ask how one may identify such individuals within a given population. One possible group would be sex workers. However, the prevalence of gonorrhea amongst sex workers in the United Kingdom has been found to be less than 2%,^{26} which is not higher than that in at-risk youth.^{18} Another possibility would be individuals involved in concurrent partnerships. While not explicitly modeled in our framework, concurrent partnerships can be viewed as a series of “rapidly changing monogamous interactions.”^{27} Indeed, one study which interpreted gap lengths as the time between a run of sexual encounters with any 2 consecutive partners, showed that gaps preceding sex with known partners were shorter than gaps leading up to sex with new partners (mean of 20.6 days vs. 7.9 days, respectively).^{28} The sequential short gaps required for persistence may thus be occurring in the framework of concurrent partnerships. Other mechanisms for facilitating the pairing of individuals with consecutive short gaps would be the existence of common meeting places for individuals seeking partnerships following short gaps or the existence of subpopulations with a much higher prevalence of individuals having the appropriate sequence of gap-partnership lengths. Such behavior has been found to be reasonably common in some studies of young people,^{19,23} and younger age was associated with shorter partnership and shorter gap lengths within NATSAL data. The higher prevalence of such behavior in youth may explain the high burden of gonococcal disease in this age group.^{1}

When used to look at potential interventions, our model gives somewhat different insights from previous work using the classic model.^{6} Behavioral changes reducing single day partnerships may have no beneficial effect on gonococcal transmission, while the effect of reducing short and midterm partnerships is modest; a far more substantial effect occurs by reducing the proportion with short gaps between partnerships. Change in behavior towards longer partnerships is thus likely to be less effective than change in behavior that prolongs the period between consecutive sexual partners. Furthermore, while increasing condom use was beneficial, the effect was modest unless applied at a sufficiently high level across the critical partnership lengths. In particular the key partnerships may not be the single day encounters, and health education messages must therefore also be targeted towards increasing condom use in the short to midterm partnerships of several weeks or longer.

Our model has several limitations. Firstly, while the model reasonably replicates the observed epidemiology of gonorrhea in UK men, it performed poorly in reproducing the observed pattern of infections for females.^{13} The likely reason for this is that we made the simplifying assumptions of using the same gap and partnership distributions for men and women and excluded individuals reporting concurrent partnerships. Men appear more likely to have concurrent partnerships than women^{11} and this could explain why the observed proportion of women reporting long-term partners as the source of their infection^{13} is much higher than that predicted by the model. Secondly, we assumed the same baseline condom use regardless of partnership length, which may be overly simplistic. However, we did not observe any difference in condom use by partnership length, except in respondents who reported partnerships comprising only 1 sexual encounter (where it was 55% compared to 18% in those reporting longer partnerships). Finally, there was considerable uncertainty in our estimates of the proportions in different partnership and gap lengths, particularly due to the amount of missing data for the latter. In addition, the way in which individuals with different gap lengths form partnerships and the degree to which individuals change their gap and partnership behavior after each relationship cycle is not known. The degree of cross-over (ρ) had a greater impact than mixing assumptions (ε), and for gonorrhea to persist, we had to assume a very low value for ρ, implying that individuals seldom crossed gap-partnership cycles. However, this may be realistic given that individuals who are part of a chain of transmission for gonorrhea must have at least 2 gap-partnership cycles of the appropriate lengths – one cycle within which to acquire the infection and another within which to pass on the infection. Thus a gonococcal chain of transmission is essentially sustained by the assortative mixing of individuals with at least 2 consecutive cycles of short gaps and short to midlength partnerships.

In summary, our results suggest that whilst there is assortative mixing of a pool of individuals who sequentially have short gaps between partnerships of the appropriate length, there is likely to be a niche for the transmission of gonorrhea. This group does not need to be necessarily the individuals with the highest partner change rates, as the partnership lengths most efficient for the transmission of gonorrhea may not be the shortest partnerships. To date, in empirical studies, risk behavior has largely been measured in terms of numbers of sexual partners. In contrast, our results suggest that of equal or greater importance is the gap between partnerships and future research on sexual risk behavior needs to “mind the gap.”

## References

## Appendix

Let *Mijl* and *Fikm* denote single males and females and *Pijklm* denote those in partnerships, with each individual being assigned to a designated gap-partnership length combination. For singles, the first array, *i*, gives the designated partnership length, and the second (*j* in males, and *k* in females) the current gap length. For those in partnerships, *i* likewise gives the designated partnership length, but *j* and *k* denote the preceding gap lengths for the male and female members of the pair. In addition, we tracked 3 infection states (subscript *l* in males and *m* in females), where 1 = symptomatic care-seeking, 2 = uninfected, and 3 = infected but noncare-seeking. Rate equations describe transitions between paired and single states and between infection states. For paired individuals,

where

describes partnership dissolution, *EijklmP* describes partnership formation, *HijklmP* describes recovery from infection, and *Iijklm* describes acquisition of infection. Acquisition of infection does not occur amongst singles, so that, in single males and females,

where

and

describe singles moving into partnerships, *EijlM* and *EikmF* are singles leaving partnerships, and *HijlM* and *HikmF* describe recovery from infection.

Let *Bijk* be the entry rate into each gap-partnership subcompartment of the partnerships:

when *j* = *k* (i.e., males and females from the same preceding gap length), and

when *j* ≠ *k* (i.e., males and females are of different gap lengths), where *Sij* and *Sik* are the number of single males and females and ε is the mixing parameter (ε = 1 represents fully assortative mixing and ε = 0 fully proportionate mixing).

The proportion of different infection state combinations entering partnerships, *Cijklm*, is the product of the prevalence of infection states within single males and females:

The entry rate of different infection states and gap-partnership combinations into pairs is thus:

Recovery from infection redistributes infection state combinations, so that *HijklmP* is described for different combinations as:

Likewise, acquisition of infection, *Iijklm*, is described as:

where θ*M* and θ*F* are the proportions of incident infections which are symptomatic and care-seeking, and π*iM* and π*iF* are probabilities for acquiring the infection per unit time in males and females, respectively.

π*iM* and π*iF* are specific to the partnership category, being not just related to the probability of disease acquisition per sex act (β*M* and β*F*), but also the product of the proportion which did not consistently use condoms (1 − χ*i*), *and the frequency of sexual intercourse within partnerships, ζi*:

where σ*M* and σ*F* are the average duration of infectiousness for symptomatic care-seeking males and females, and κ is the corresponding duration for noncare-seeking individuals.

For the rate equations for singles (equations.[ 2] and [3]), let the number from each infection state and gap-partnership combination which leave the paired state be:

The sum of all the males and females of each infection state leaving partnerships would be:

Since we assume the distribution between singles and partnerships is constant, the entry rate into singles within each gap-partnership cycle corresponds to the rate of partnership formation, i.e., *Sij*/φ*j*. The composition of infection states is determined by the proportion coming from the original gap-partnership cycle (1−ρ), and the proportion coming from individuals crossing over from other gap-partnership combinations, ρ, as follows:

Recovery of single individuals results in redistribution between infection states, so that: