2˳ Influence of sexual contact patterns on epidemiology of sexually transmitted infections
From an epidemiological point of view, the transmission dynamics of STIs are fundamentally different to those of many other infectious diseases˳ First, sexual contact rates are usually invariant to population size, which means that there is no critical population density required for a typical STI to persist˳ By contrast, the rate at which non-STIs spread is often dependent on the density of the host population [8]˳ Second, there is often considerable variation in sexual behaviour both within [4] and between [9] populations˳ Highly active members of the population (e˳g˳ sex workers in human populations, alpha males in animal populations) will generally be at much greater risk of receiving and transmitting an infection than monogamous couples and so contribute disproportionately to the spread of disease as well as representing important targets for disease control˳
In order to establish how heterogeneity in sexual behaviour can alter epidemiological dynamics, it is first useful to define the basic reproductive number, R0, of an infectious agent in a well-mixed population:
where β is the probability of transmission per contact, D the average infectious period and n the average number of contacts (see [8] for a more detailed discussion of R0)˳ The basic reproductive number is essentially the average number of secondary infections that a single infectious individual will produce in an entirely susceptible population˳ Hence the infection will tend to spread if R0 > 1, but will go extinct if R0 < 1˳ This formulation of R0 is based on an idealized, randomly mixing homogeneous population, but real mixing patterns are likely to be more complex due to spatial constraints and variations in host behaviour˳ If we imagine a continuum with well-mixed and highly structured populations at the extremes, then most real populations will fall somewhere between the two˳ Note that the position of a population on this continuum is dependent on the transmission pathways of a particular infection; a population may be relatively well-mixed in terms of social contacts, but might demonstrate a high degree of heterogeneity in sexual mixing patterns˳
In general, casual contacts between humans tend to be ephemeral and non-repetitive, whereas sexual contacts are more stable [10]˳ In addition, variation in close contact rates is likely to be much smaller than variation in sexual partner acquisition rates˳ For example, Mossong et al˳ [11] found that adolescents had just over twice the number of close contacts than the elderly, but studies of sexual mixing patterns generally find power-law distributions in partner acquisition rates, sometimes ranging over three orders of magnitude [4,12]˳ Power-law distributions are also likely to be applicable to a variety of animal mating systems, particularly where a few members of one sex are dominant (i˳e˳ polyandry or polygyny)˳ Hence, the above formulation of R0 may be a reasonably good indicator of epidemic spread for infections transmitted by close contact, but is likely to be a poor approximation for STIs˳ In addition, sexual contacts tend to be much less frequent than social contacts, lowering the value of R0˳ Hence, one explanation as to why many STIs are associated with chronic, asymptomatic diseases is that this increases the value of D to compensate for lower contact rates˳
Sexual transmission can be considered part of a much broader class of models with heterogeneous contact rates, usually referred to as ‘super-spreader’ models, where a few members of the population have a disproportionately large effect on disease spread [8,12]˳ For super-spreader models, we can incorporate this heterogeneity into the formula for R0 by compartmentalizing the population according to contact rates, so that Ni is the proportion of the population that acquires i contacts per unit time [13,14]˳ Retaining the assumption that mixing is random, this allows us to calculate an effective contact rate, c, over the distribution:
where m and σ2 are the mean and variance in contact rates, respectively [14]˳ The formula for the basic reproductive number now becomes R0 = βDc; clearly, any heterogeneity in contact rates will increase the value of R0 and hence the initial growth rate of the epidemic (figure 1)˳
In the context of sexual transmission, the second formulation of R0 applies to a homosexual population, but it can be readily generalized for a heterosexual population by separating the effective partner acquisition (i˳e˳ contact) rate into male (cm) and female (cf) components˳ If we also assume that there are differential transmission rates across the sexes (as is common with many STIs), then our equation for R0 becomes:
where βm is the transmission rate from males to females and βf is the transmission rate from females to males [8,16]˳ If cm = cf then R0 will asymptote towards quadratic growth with the coefficient of variation (CV = σ/m), but if variation is limited to one sex then R0 will tend towards linear growth˳ Changes in the variance will be most significant when R0 is close to unity (the epidemic threshold), as relatively small changes in the size or the behaviour of the core group can determine whether an epidemic will occur (figure 1)˳
The effective partner acquisition rates can also be used to estimate the ratio of cases in males (Cm) to females (Cf) during the early stages of the epidemic
(see [8], §11˳3˳9 for a more detailed discussion; also [15])˳ This work was originally motivated by the spread of HIV in Africa, but the principles can be applied to other populations that exhibit host heterogeneity˳ For example, the ratio Cm/Cf suggests that the dominant sex in a biased mating system (e˳g˳ males in polygynous systems) will tend to exhibit lower than average levels of infection, although this could be counterbalanced by differences in transmission probabilities˳ Indeed, it is thought that Cm/Cf ≈ 1 for HIV-1 in many parts of Africa because the partner acquisition rates (cf > cm) are more or less balanced by differences in transmission rates (βf < βm) [8,16]˳ Note that even if Cm/Cf ≈ 1, the distribution of infection will still be biased towards more sexually active members of the population˳
Further complications will arise if the population does not mix homogeneously, for example where people tend to show a preference for mixing with similar individuals (assortativity)˳ Mixing patterns have been found to vary considerably between human populations, ranging from highly assortative [17], to highly disassortative (i˳e˳ showing preference for dissimilar individuals) [18] mixing˳ The degree of assortative mixing may also vary within a population: for example, Wylie & Jolie [19] found that assortative mixing was common in linear components of a sexual contact network (SCN), but disassortative mixing was common in radial components˳
In order to model heterogeneous mixing, we can group individuals according to their level of sexual activity (i˳e˳ partner acquisition rate) and describe interactions between groups using a ‘mixing-matrix’ [13,14,16,20,21]˳ A simple mixing-matrix for a population split into high (H) and low (L) activity groups would be
where pij is the proportion of sexual contacts that individuals from group i make with members of group j˳ For completely assortative mixing, pij is equal to the identity matrix (pii = 1, pij = 0 for i≠j)˳ There is usually no single disassortative extreme, however, as disassortativity is maximized whenever the elements of the main diagonal of pij are minimized [21]˳ For a given mixing matrix, we can measure the degree of assortativity, Q, in the population as
where g is the number of activity groups and λi are the eigenvalues of pij˳ Gupta et al˳ [21] found that highly assortative mixing (Q ≈ 1) tends to lead to more rapid epidemic growth and can produce multiple peaks in disease incidence˳ By contrast, highly disassortative mixing (Q ≈ −1/(g − 1)) is generally associated with slower epidemic growth, but will typically produce higher peaks in disease incidence (figure 2)˳ This method highlights the importance of host heterogeneity in the spread of STIs and suggests that targeting control measures at the core group is optimal, although the efficacy of such procedures will depend on the size of this group and the degree of assortative mixing in the population˳
While this approach is a useful way of capturing host heterogeneity, it cannot capture some of the complex interactions found in real populations that are imposed by other factors than level of sexual activity˳ Such mean-field approaches assume that sexual activity classes are well mixed so that if an infectious individual mixes with a particular activity class, then all members of that class will have an equally increased risk of infection˳ In reality, the risk of infection will be limited to those who have sexual contact with the infectious individual rather than the entire activity class˳ An alternative method is to use an SCN which captures heterogeneity at the level of individuals and provides a means of replicating more realistic transmission pathways˳ This approach is particularly well suited to STIs, as transmission pathways are usually much more clearly defined (i˳e˳ sexual contact) than for non-STIs˳ However, there are many problems associated with collecting data on real SCNs, including biases in reporting and difficulties with linking up components in a larger network [17,22], although some attempts have been made for small populations [18,19,23-26]˳
Mean-field models and SCNs may show good agreement over the main part of an epidemic, but fundamental differences in structure are likely to have significant consequences for the spread of infection during the early stages [27]˳ Keeling [27] showed that in a general SCN, the probability that an index case i will fail to pass on the infection to any of their ni contacts is
where R0(i) = min(βDni,n) is the expected number of secondary infections to be caused by the index case˳ The corresponding probability of extinction after the first generation using a mean-field approximation is:
which satisfies PMF(i) > PSCN(i)˳ Hence the probability of extinction during the first generation is always higher in a mean-field approximation than it is in the corresponding SCN, but as the number of contacts increases the two values converge (i˳e˳ )˳ Averaging these values over the entire population gives the probability that a randomly introduced infection will die out after the first generation˳ Note that in real populations, an index case is probably more likely to occur among highly active individuals, which will decrease the probability of extinction during the first generation˳
One advantage that SCNs have over mean-field approximations is their ability to capture long-term partnerships that are commonly found in many human and animal populations˳ In particular, serially monogamous partnerships (common among birds as well as humans) cannot be modelled using traditional mean-field approaches˳ Computer-generated contact networks can be used to recreate mixing patterns observed in real populations [25], by connecting individuals (nodes) to other members of the population preferentially, based on factors such as proximity, cluster size or assortativity˳ Studies of simulated epidemics on SCNs have revealed that concurrent partnerships are crucially important to the spread of many STIs [28-30]˳ For example, Morris & Kretzschmar [29] demonstrated that the size of an epidemic grows exponentially with the relative number of concurrent partnerships in a population˳ Reducing the number of concurrent partnerships in a population is therefore likely to be an effective mechanism of disease control˳
