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Number of Sexual Encounters Involving Intercourse and the Transmission of Sexually Transmitted Infections

Nordvik, Monica K. BA; Liljeros, Fredrik PhD*†

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Sexually Transmitted Diseases: June 2006 - Volume 33 - Issue 6 - p 342-349
doi: 10.1097/01.olq.0000194601.25488.b8
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SEXUALLY TRANSMITTED INFECTIONS (STIs) are a great problem in many Western countries today. Chlamydia, gonorrhea, and human immunodeficiency virus (HIV) are endemic, if not epidemic, to many places, and efforts are under way to understand the dynamics of the spread of such infections. Much of the focus has been on understanding what kind of behavior can be understood to increase the probability of contracting and spreading these infections. In most epidemiologic models, it is assumed that the number of sexual partners largely determines the rate at which the number of secondary cases arises. Also, this relationship is often assumed to be linear, with secondary infections increasing proportionally with the number of partners.1

But modelers have also long been aware of that the number of sexual encounters involving intercourse per partner also has the potential to affect the number of secondary cases,1 and recently it was suggested by2 that the relationship between the number of sexual partners an individual has and the number of individuals he or she eventually infects may not be linear.

In this paper, we will proceed from the finding that this relationship weakens when we take into account the number of sexual encounters involving intercourse with each partner in question, and is, indeed, no longer linear. By using Swedish survey data on sexual behavior, we will show that when we model the heterogeneity in terms of number of sexual encounters involving intercourse per partner, the basic reproductive rate, R0, is significantly lower than when homogeneity is assumed and, finally, we will examine the magnitude of this bias.

Background

The basic reproductive rate, R0, is a widely used measure of the spread of infectious diseases and the risk of epidemic outbreak. The parameter is usually defined as “the average number of secondary cases caused by an infectious individual in a completely susceptible population.”1 Once the disease has entered the population, one can no longer assume that all individuals are susceptible. It is then appropriate to talk about the reproductive rate for each new generation of infected individuals, Rg, a measure of which the basic reproductive rate is a special case.

The basic reproductive rate characterizes the epidemic threshold in such a way that if R0 >1 the disease will spread in the population and an epidemic is possible, if R0 <1 the disease is expected to disappear, and if the parameter equals 1, the disease is endemic to the population.1

R0 is affected by both the epidemiologic characteristics of the disease itself (transmissibility, duration of infectious period) and the population (number and pattern of contacts between individuals). The basic reproductive rate, R0, is therefore in its simplest form defined as:

where z is the mean number of contacts per unit of time, β is the probability of transmission per contact, and τ the mean duration of the infectious period.1

When using this model, one assumes that all susceptible individuals have the same probability of becoming infected, that is, that there is no risk structure. However, a number of studies have shown that for diseases where a risk structure exists, which is the case with STIs (see, for example, Laumann et al.3), it is important to take this into account.4 One example is that, given that transmissibility is homogeneous for all relationships, individuals with a larger number of partners are assumed to have a relatively higher risk of becoming infected, and if infected, they are also assumed to have a higher probability of infecting a larger number of people. What happens is that for epidemiologic purposes, the mean number of partners is underestimated when homogeneity is assumed. This can lead to underestimating R0, which can be corrected for by taking the variance in the number of partners into account. The corrected formula for R0 is thus:

where z is the mean number of partners per unit of time, σ2 is the variance in number of partners, β is the probability of transmission per partner, and τ the mean duration of the infectious period.1

In equation 1, R0 is a linear function of the mean number of contacts per time unit. This assumption may, however, not hold true if the number of sexual encounters involving intercourse that a person has during a period of time is constant or negatively correlated with the individual’s number of partners (see also Blower and Boe7). In Figure 1, we show how the number of secondary cases changes when we keep the total number of sexual encounters for each person fixed, but let the number of partners per person vary (equation shown in Appendix A). This assumption is based on the reasonable hypothesis that it is at least as easy for an individual with a stable partner to have a large number of sexual encounters involving intercourse with that partner as it is for an individual with a larger number of partners to have additional sexual encounters with one of these partners. It requires much more effort to initiate sexual contact with a nonstable partner per encounter than to initiate a sexual encounter with a stable partner.2 The distribution of the total number of partners and total number of sexual encounters involving intercourse in the data used in this paper supports this assumption and is shown in Figure 3.

Fig. 1
Fig. 1:
Total number of secondary cases per total number of partners (ψ = probability of transmission per act of sexual intercourse).
Fig. 3
Fig. 3:
Total number of sexual encounters involving intercourse per total number of partners. A, Women. B, Men (outliers marked are singular cases).

Figure 1 shows that the relationship between the number of sexual partners that an infectious individual has and the number of individuals he or she eventually infects is not linear. The increase in the number of secondary cases per additional partner is larger for individuals with a small number of partners than for individuals with a large number of partners. This pattern is especially strong for less contagious diseases, that is, for diseases with a low probability of transmission per sexual intercourse (see also Garnett8). This is explained by the fact that the addition of a new partner will decrease the number of sexual encounters involving intercourse that the individual has with each partner as reasoned above, which in turn will decrease the probability of transmission nonlinearly. As we can see in Figure 1, the frequency of sexual intercourse affects the number of secondary cases when the probability of infection per sexual intercourse is as high as 0.5. This indicates that the number of sexual encounters could be an important factor both for less contagious diseases such as HIV9 and also for highly contagious infections such as gonorrhea, were the probability for female-to-male transmission has been reported by Yorke et al.10 to be 0.5.

The fact that there are differences in frequency of sexual encounters involving intercourse between different relationships that may affect the number of secondary cases produced by every individual has also been pointed out by Anderson and May,1 among others. One way that is used by modelers to handle this problem is to apply different transmission probabilities for different types of relationships: lower probability for casual encounters and higher for steady relationships (see, for example, Kretzschmar et al.,11 Oxman et al.,12 Bauch13). This way of modeling the complex relationship between transmission probabilities and number of partners or frequency of sexual encounters involving intercourse has been evaluated by Kaplan14 and was found more reasonable than asserting a probability of transmission for each sexual intercourse, at least when it comes to some STIs; for example, HIV.

The bias that arises when one assumes that different kinds of relationships carry with them the same probabilities of transmitting and contracting an STI has, however, not been exhaustively studied, and the main objective of this paper is therefore to estimate the direction and the magnitude of this. We will therefore take both the number of sexual partners an individual has and the frequency of sexual encounters involving intercourse into account when calculating the basic reproductive rate, R0. We will use empirical data to show that when we ignore this heterogeneity in the individual’s behavior, but consider only the mean number of partners and mean number of sexual encounters involving intercourse per partner, R0 (using equation 1) can be overestimated (in contrast to the case described in May and Anderson5).

Research on contact networks has also shown that there are factors other than the absolute number of partners—such as network position15,16 and level of concurrency of sexual relationships17–19—that affect an individual’s risk of contracting and spreading an STI. These factors will, however, not be included in this article because we have chosen to focus solely on the number of partners and frequency of encounters.

Materials and Methods

Data

The data used in this paper come from a survey that was conducted in Sweden in 1988. The reason for using such relatively old data is that they appear to be the only data available that were collected by a random sample of a general population (in contrast to Blower and Boe7) and include questions on both the number of partners and the number of sexual encounters involving intercourse per partner, without grouping any of the variables (which was done in Laumann et al.3). A random sample of 1150 individuals aged 16 to 31 years was drawn (10% of the population in that age group) on Gotland, an island in the Baltic Sea that has an age and sex distribution similar to that of Sweden as a whole. The response rate was 68% (768 individuals), and it was concluded that the material was free of systematic biases20 such as sampling biases or dropouts related to specific groups of individuals.

The respondents were asked to come to different schools or similar institutions to fill out an anonymous self-administered questionnaire. Included were questions on age of first intercourse, age of intercourse with second partner, number of lifetime partners, and sexual behavior over the past 12 months.

One unique aspect of this survey is that the respondents were asked not just to indicate the number of partners during the year preceding the survey but also when during the year the relationship took place, the duration of the relationship, and the approximate number of sexual encounters involving intercourse with each partner (the questionnaire is shown in Appendix B). This not only enables us to model the possible spread of a STI in relation to the number of partners but also takes into account the frequency of the contacts.

Figure 2 shows the proportion of men and women, respectively, with different numbers of partners during the year. As we can see, the distribution is positively skewed and the variation is very high, with most of the men and women having only 1 or 2 partners during the year. Figure 3 shows the total number of sexual encounters in relation to the total number of partners in the data. The assumption about linearity in this relationship is reasonable; in fact, it seems that for women, the total number of encounters decreases rather than increases with number of partners.

Fig. 2
Fig. 2:
Percent of men and women with different number of partners.

Calculations

For the sake of simplicity, we set the time unit to 1 in both of our calculations; that is, we assumed that the duration and infectiousness of the disease in question are present and constant during the whole year. This may not be correct for bacterial infections such as chlamydia (transmissibility unknown, probably episodic) or gonorrhea, where transmissibility may extend for months if untreated. However, it is probably reasonable to assume for HIV, which is presumably transmissible over the entire lifetime of the individual, although the level of transmissibility may vary over time.21 For simplicity, we also assume that transmissibility is homogeneous for both sexes, and we do not account for any risk factors except the number of partners and the number of sexual encounters involving intercourse per partner.

In our first case, we assume that the number of sexual encounters is homogenous in the population (that is, we base our calculations on the overall mean numbers of partners and encounters per partner). We therefore define R0 by:

where ψ is the probability of transmission per sexual encounter, y is the mean number of encounters per partner, and z is the mean number of partners.3

From the data, we calculate the mean number of partners for women to be 1.5 and for men 1.7.4 In most research on sexual behavior, there is a discrepancy between the number of partners reported by women and men, and the reasons for this have been discussed.22,24 However, the discrepancy in this data is found to be low in this data compared to the findings of many other studies.3,24 We also calculate the mean number of sexual encounters involving intercourse per partner to be 45.6 for women and 35.5 for men. We then vary the transmissibility of the disease to see how R0 behaves under these conditions.

In our second case, we take into account some of the heterogeneity in the population. This calculation is therefore based on the actual number of partners for each individual and the actual number of sexual encounters involving intercourse per specific partner. R0 is consequently defined as the mean of all the individuals’ expected number of secondary cases, Rindividual, which is calculated as follows:

where i is the partner, m is the total number of partners for that individual, n is the total number of individuals in the population, ψ is the probability of transmission per sexual encounter, and y is the number of sexual encounters with the partner in question.

Results

Assuming Homogeneity

In our first case, we based our calculations on the mean number of partners and the mean number of sexual encounters involving intercourse per partner, as discussed above. As is shown in Figure 4,5 the basic reproductive rate quickly reaches and exceeds 1; that is, an epidemic is possible even when the transmissibility of the infection (ψ) is very close to zero. For women, R0 reaches 1 at a slightly lower value of ψ (between 0.025 and 0.026) than for men (ψ = 0.026). This is because the women in the study had a lower mean number of partners (1.5) and a higher mean number of encounters with each partner (45.6) than did the men (1.7 partners and 35.5 sexual encounters per partner). These differences in the value of R0 are, however, not statistically significant.

Fig. 4
Fig. 4:
R0, calculations based on the mean values of number of partners and mean number of sexual encounters involving intercourse per partner (assuming homogeneity, Eq. 3). By sex (dotted lines = 95% CI obtained by bootstrapping).

The parameter also quite quickly converges on a state with a stable value. For women, R0 stabilizes at 1.45 when ψ = 0.15, and for men the analogous numbers are R0 = 1.65 and ψ = 0.2. One interesting thing is that when transmissibility is lower than 0.03, R0 is larger for the women than for the men, but when ψ is larger, the men tend to infect a larger number of partners (not statistically significant). This occurs because the women have a larger number of encounters per partner, and this affects the probability of transmission most when the transmissibility is low and decreases as ψ increases.

Modeling Heterogeneity

When we instead first calculate each individual’s expected number of secondary infections and then their mean value as also discussed above, the result changes, as shown in Figure 5. It reaches and exceeds 1 for both sexes at a higher value of ψ than in the example above; only when ψ is between 0.14 and 0.15 for women and 0.12 and 0.13 for men does the value of R0 reach 1. The differences between men and women are, however, not statistically significant.

Fig. 5
Fig. 5:
R0, mean value of each individual expected number of secondary cases (modeling heterogeneity, Eq. 4). By sex (dotted lines = 95% CI obtained by bootstrapping).

In this second case, the value of R0 never stabilizes but continues to grow over the entire range of ψ. This is very interesting because it means that R0 is always affected by the transmissibility of the disease and therefore also by the number of sexual encounters involving intercourse. The effect is large when transmissibility is low, and it decreases as ψ increases, which is also manifested by the fact that (in this case as well as in the first) the women have a higher R0 than men, though not statistically significant, at a low value of ψ (under ψ = 0.1) than at a higher one.

In Figure 6, we compare the 2 ways of calculating the value of R0. As indicated, the differences between the values of R0 when assuming homogeneity and when modeling heterogeneity are statistically significant up to ψ = 0.63, where both parameters converge.

Fig. 6
Fig. 6:
Comparison of the parameters. Red line: assuming homogeneity in number of sexual intercourses per partner; black line: modeling the heterogeneity in the data (dotted lines = 95% CI obtained by bootstrapping).

Discussion

When modeling the spread of STIs, one of the most important factors to take into account is the interaction structure between infected and noninfected individuals. This paper focuses on the relationship between an individual’s number of sexual partners, frequency of sexual encounters involving intercourse with each partner, and the number of partners to whom he or she eventually transmits the infection.

As mentioned earlier, it has been asserted that the mean number of sexual contacts is often underestimated in epidemiologic studies and that the reason for this is that they assume homogeneity in the number of partners. This can lead to underestimating R0, which can be corrected for by taking the variance in number of partners into account (Eq. 2).1

It has also been pointed out that there is a difference in the frequency of sexual encounters involving intercourse between different kinds of relationships.7 If an individual has a large number of partners, it is reasonable to think that he or she does not have as many sexual encounters with each partner as someone who only has a few. This heterogeneity may also affect the transmission of STIs.1 These issues have been managed by some modelers by applying different transmission probabilities for different types of relationships (for example, see Kretzschmar et al.,11 Oxman et al.,12 Bauch13), something that has been evaluated and found to be a more reliable way of calculating the spread of STIs than doing the same for each sexual encounter.14

So it is clear that the fact that there is a potential bias related to assuming homogeneity in the number of sexual encounters involving intercourse in different types of relationship has been known. But the magnitude and the direction of this bias have, however, not been analyzed.

In this paper, we have shown that the basic reproductive rate, R0, tends to be overestimated over the entire range of ψ (probability of transmission) when not taking this heterogeneity into account. This is because most individuals have a smaller number of sexual intercourses with casual, nonstable partners than with steady ones. This heterogeneity can, of course, lead to either under- or overestimation of R0, depending on how the distribution of number of partners looks. Various studies3,24,25 have, however, shown that the variation in an individual’s partners is very high and seems to be skewed in the same way as in our case with most of the individuals only having a small number of partners (Fig. 2), which implies that the basic reproductive rate tends to be overestimated in the way that is shown here.

These results indicate that it is possible that individuals who have a large number of partners may not, as is often assumed, be alone in playing a central role as spreaders of STIs. It may, instead, be that individuals who have a large number of sexual encounters involving intercourse per partner and have several, but not necessarily a very large number of, partners also play a great role in the transmission of STIs. This is, of course, only true if the transmissibility of the disease is fairly low; otherwise, the profit to the STI of multiple opportunities of infecting the same individual would only be marginal.8 The difference between the most effective way of spreading different STIs—by means of a large number of encounters or a large number of partners—may help us to understand why the increases in infections like chlamydia or gonorrhea, which both have a high transmissibility and are almost epidemic in many countries, have not been followed by similar increases in STIs with low transmissibility such as, for example, HIV. The behavioral change from a small to a large number of partners benefits many infectious diseases, but not those at the lower end of the scale in terms of probability of transmission per sexual encounter.

What happens to the effect of the frequency of sexual encounters when other important factors are considered, including changes in the number of partners, different transmissibility for sexes, network position, and concurrency of sexual relationships, is an open question that needs to be explored further. One approach might be to use the theory of dynamic weighted networks, an approach that would make it possible to capture both the frequency of the encounters and concurrency of sexual relationships, while at the same time modeling the fluctuations in these factors.

The outcomes of this paper suggest that it is not only the number of partners that is important for the transmission of disease but also the characteristics of the contacts with the partners in question. Frequent contact (with many sexual encounters involving intercourse) increases the risk of infecting the partner, and transmissibility is incorrectly calculated when the heterogeneity in this variable is not taken into account. But other modelers have also been questioning this approach and arguing that a constant transmissibility per relationship can be as misleading as constant per-act transmissibility. As an example, when examining the relationship between these parameters, Røttingen and Garnett26 found that either the probability of transmission of HIV is extremely diverse in different relationships or it actually decreases over time. This suggests that there are other characteristics in the relationships that also affect the transmission of these diseases, something that clearly shows that we need more empirical studies of transmission of STIs to enhance our understanding of these dynamics and to be able to make models that are as accurate as possible.

References

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    Appendix A

    The relationship between R0 and number of partners per year, k, was calculated according to:

    where ψ is the probability for transmission of an STI per sexual intercourse and Y the total number of sexual contacts a year. In the expression, the number of sexual encounters involving intercourse is assumed to be independent of number of sexual partners.

    Appendix B: Questionnaire

    Please note that this is a translated description of the questionnaire and not an exact copy.

    Instructions

    Assign a number for each of your partners during the last year. Mark your longer relationships with a line for the entire time the relationship lasted and 1-time contacts with an X on the approximate date they occurred.

    Then fill in the following information about each of the partners:

    • (A) Age of the partner (approx.)
    • (B) Number of sexual encounters involving intercourse (approximate if you don’t remember; try to remember how often or how many times you met and then how often you had intercourse)
    • (C) Did your partner live on Gotland?
    • (D) Did you use a condom? (always/never/approximately how often)

    Example:

    Table
    Table
    Table
    Table
    Table
    Table

    Appendix C: Bootstrapping

    For accuracy, many of the statistical estimates that we use require that the underlying distribution be assumed to be normal. Bootstrapping is a nonparametric procedure using resampling that is used to estimate different parameters when this requirement is not met, such as when we have either a small sample or the sampling distribution is unknown.

    The procedure goes as follows: if we have a sample of size n, we draw a large number of resamples (at least 1000 is recommended), with replacements, of size n from our sample. The relative frequency distribution of these values is used as an estimate of the underlying sampling distribution.

    In order to obtain a confidence interval with a 0.05 α level, we simply use the 2.5th and the 97.5th percentile in the estimated sampling distribution as endpoints. If the number of resamples was 1000, this would correspond to counting up to the 25th lowest value and counting down to the 25th highest value, and we would obtain the endpoints for the confidence interval we wanted.

    Mooney CZ, Duval RD. Bootstrapping. A Nonparametric Approach to Statistical Inference. Newbury Park: SAGE Publications Inc, 1993.

    1Contact here means any type of contact that has the potential to lead to a transmission of the infection.
    Cited Here

    2One possible exception to this last assumption is contact with prostitutes, where the opposite may be true. Limited economic resources are, however, likely to put an upper limit to the number of such encounters.
    Cited Here

    3When we derive this formula, and the following, we use the fact that the probability of not being infected by a partner can be written as (1 − ψ) powered to the number of intercourses, and that the probability of being infected at least once can therefore be written as [1 − (1 − ψ)y].
    Cited Here

    4Because the distribution of the number of partners is highly skewed (see Fig. 2), it may be more appropriate to use the median; however, we chose to use the mean because the formula by [5] is based on this parameter.
    Cited Here

    5For a description of the bootstrapping procedure, see Appendix C.
    Cited Here

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