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Conceived and designed the experiments: LECR FL PH. Performed the experiments: LECR PH. Analyzed the data: LECR FL PH. Wrote the paper: LECR FL PH.

The authors have declared that no competing interests exist.

Sexual contact patterns, both in their temporal and network structure, can influence the spread of sexually transmitted infections (STI). Most previous literature has focused on effects of network topology; few studies have addressed the role of temporal structure. We simulate disease spread using SI and SIR models on an empirical temporal network of sexual contacts in high-end prostitution. We compare these results with several other approaches, including randomization of the data, classic mean-field approaches, and static network simulations. We observe that epidemic dynamics in this contact structure have well-defined, rather high epidemic thresholds. Temporal effects create a broad distribution of outbreak sizes, even if the per-contact transmission probability is taken to its hypothetical maximum of 100%. In general, we conclude that the temporal correlations of our network accelerate outbreaks, especially in the early phase of the epidemics, while the network topology (apart from the contact-rate distribution) slows them down. We find that the temporal correlations of sexual contacts can significantly change simulated outbreaks in a large empirical sexual network. Thus, temporal structures are needed alongside network topology to fully understand the spread of STIs. On a side note, our simulations further suggest that the specific type of commercial sex we investigate is not a reservoir of major importance for HIV.

Human sexual contacts form a spatiotemporal network—the underlying structure over which sexually transmitted infections (STI) spread. By understanding the structure of this system we can better understand the dynamics of STIs. So far, there has been much focus on the static network structure of sexual contacts. In this paper, we extend this approach and also address temporal effects in a special type of sexual network—that of Internet-mediated prostitution. We analyze reported sexual contacts, probably the largest record of such, from a Brazilian Internet community where sex buyers rate their encounters with escorts. First, we thoroughly investigated disease spread in this dynamic sexual network. We found that the temporal correlations in this system would accelerate disease spread, especially at shorter time scales, whereas geographical effects would slow down an outbreak. More specifically, we found that this contact structure could sustain more contagious diseases, like human papillomavirus, but not HIV. These results highlight the importance of prostitution in the global dynamics of STIs.

Spatiotemporal heterogeneities in sexual contact patterns are thought to influence the spread of sexually transmitted infections (STIs). Since epidemics can be a society-wide phenomenon, and sexual contact patterns can have structure at all scales, we need population-level sexual network data to understand STI epidemics. Unfortunately, it is hard to collect sexual contact data on that large a scale. Instead, people have focused on small-scale studies using interviews

Sexual contact patterns have temporal correlations both at an individual and at a population level

Extending epidemiological models to include space is a common step towards inclusion of structure beyond the well-mixed assumption

In this paper, we address the question of how the dynamic contact structure in the contact data of Rocha

The web community from which our dataset is obtained is a public online forum openly visible online. The full dataset is available as support information (

Apart from the anonymous aliases of sellers and buyers and time stamps, posts also include the buyers' grades of the escorts' performance and information about the types of sexual activity performed during an encounter, divided into three categories: oral sex (with or without condom), mouth kissing, and anal sex. All posts, however, are assumed to report vaginal intercourse (random inspection supports this assumption). In our simulations, for the sake of simplicity, unless otherwise stated, we use all available links and disregard the fact that they possess different levels of risk. Most contacts between a seller and buyer happen only once. By inspection, several users report that next time they buy sex, they prefer a different escort, even if the encounter was graded good. We can expect that not all Brazilian escorts and customers of such are present in the data. Furthermore, posting about an encounter is a low-cost action by the sex-buyer that gives him status in the community, which makes it likely that the reports from most users are quite complete. For most of this paper, we ignore this and study disease spread on a network defined by our data set as it is, which limits our conclusions to effects of temporal structure relative to various other scenarios.

A common method of studying correlations in empirical contact data is to compare a network with ensembles, where some properties (like the number of nodes and their degrees) are kept constant and the rest is randomized. In the randomized network versions used in this paper, we conserve the bi-partite structure of the heterosexual network and the number of contacts of each individual.

Diverse network structures can affect disease spread

To put our results in the context of other levels of epidemiological modeling, we also consider two other contact models—a static network approach and the dynamic network model by Volz and Meyers _{active} = ^{–3}. The idea of Volz and Meyers's model is that vertices change partners with a probability _{change} while keeping the number of partners fixed over time. This model assumes that a vertex is always connected to someone else; however, in our network, in the interval of 800 days, several vertices have only one or few days during which a connection is active. This means that most of the time they are not in a position to catch a disease. To compensate for this effect, and to allow direct comparison to the simulations on the empirical network, we modify the Volz–Meyers (VM) model to capture the brevity of partnerships in the data. In our formulation of the VM network, each vertex has a chance _{k}_{k}

One can model the spread of sexual infection in various ways to capture the various characteristics of pathogens and contact patterns, and also to serve different aims of explanation and prediction. We explore the effects of temporal correlations on different levels of epidemic modeling. The first disease-transmission model we consider is the Susceptible–Infected–Removed (SIR) model. Where all individuals are initially susceptible; upon contact with an infective, a susceptible becomes infective with probability ρ (probabilities are, unless otherwise stated, per-contact probabilities), and after a fixed time δ, a susceptible changes to the removed state. If δ is larger than the vertex lifetime in the network, we get the limit case known as the Susceptible–Infected (SI) model. In a static network of finite size and non-zero transmission rate, all vertices will eventually become infected in the SI model. This is not necessarily the case in a temporal network, which makes the SI model more realistic in temporal, compared to static, contact networks.

To simulate these models in our empirical network, we first map the sampled network onto a time-ordered list. Each entry in the list is one pair of vertices and the time of the contact. Different contacts between the same pair appear as different entries in the list. Then we divide the list into intervals of 800 days each, as mentioned above. The pairs are ordered according to their times of contact. We select the sex-seller of the first contact of an interval as a source of infection and go through the ordered list infecting a susceptible vertex in contact with an infective vertex with probability ρ. The state of the vertex is updated at each new contact. A way of modeling the fact that the network is connected to a background of sexual contacts would be to include multiple sources of infection. To keep the simulations simple, however, we leave this for future studies. Since our temporal information has a resolution of one day, we do not know the order of contacts within a day. To remove this potential bias, we randomize the order of contacts within a single day 100 times. In line with other studies, and to simplify the model, we assume that both infection and removal (after time δ) are immediate, and the transmission probability is constant.

The SI model is adequate for modeling the early phase of an outbreak over shorter time scales than the duration of the disease. SIR, on the other hand, is appropriate for simulating diseases having a well-defined infectious stage followed by immunity. As an example, we will investigate HIV at a more detailed level than simply SI or SIR. Hollingsworth, Anderson, and Fraser _{1} lasting for a time _{1}, and a chronic stage of transmission probability ρ_{2}. We refer to this as SI_{1}I_{2} model. Strictly speaking, the transmissibility of HIV-1 also depends on gender and other factors such as type of sex and the fact that the viral load transmitted per-contact can spike during the chronic phase because of comorbidities, among other things. A yet more detailed model could also include an age-stratified population, as young infectives tend to influence an outbreak more. Because they are in the network for longer times, they have higher chance to establish more contacts and contribute to transmit the infection

We follow a similar procedure as above to simulate disease spread in the SN and VM networks. For the initial conditions, however, since the probability of being infected should increase with contact rate in case of the empirical networks, we now select the source of infection randomly (for each realization) and proportionally to the number of contacts of the vertex. This procedure compensates for the fact that in the empirical network, high degree nodes are necessarily selected more than once as a source of infection. This is because, on average, the chance of an individual's being active at a certain moment is proportional to that individual's number of contacts. The state of the vertex is updated after all vertices have been considered. We run the algorithm 30,000 times to obtain averages for these models.

A key quantity is the fraction of infected vertices Ω (

A straightforward way of investigating the effects of the temporal and topological structure of contact patterns is to remove different types of correlations by randomization (see Section The network models). In

In A–C, we plot the time evolution of the fraction of infected vertices 〈Ω〉. The curves correspond to SI epidemics in the original network (full line) and in its randomized versions: panel A represents swapping time stamps (RD); B shows rewiring of the edges and keeping the sellers' time correlations (RT); and panel C depicts simultaneous randomization of time stamps and edges (RDT).

The limit of high transmission probability ρ = 1 does not reflect actual STI contagion; more realistic values lie in the range 0.001≤ρ≤0.3 _{ρ}〉/〈Ω_{ρ = 1}〉, the average number of infected vertices (for probabilities ρ) relative to the number of infected vertices when the maximum transmission probability is used (ρ = 1). The relative number of infected vertices decreases within the initial 100 days and afterwards reaches a minimum for higher transmission probabilities while continuing to decrease slowly for lower rates. The minimum, which corresponds to the time lag of secondary infections, is more pronounced for lower ρ-values. The fact that the curves are fairly constant for times longer than 200 days, that is, that they converge to limiting values, is an indication that our results for the ρ = 1 case hold for other transmission probabilities as well, that is, the time-ordering effects are stronger than the fluctuations from the stochasticity of the contagion process. For lower ρ-values, the curves decrease monotonically, which indicates the existence of an epidemic threshold somewhere between, ρ = 0.01 and ρ = 0.001, which we investigate more cautiously below.

The panel shows the evolution of 〈Ω〉_{rel}, the number of infected vertices for lower transmission probabilities (0.001≤ρ≤0.3) relative to the number of infected vertices when we use the maximum transmission probability (ρ = 1). The ordinate is in log-scale.

We note that there is a large diversity of outbreaks even for ρ = 1. In

In panel A, we plot the probability distribution

To illustrate the effect of different sexual activities, we show the outbreak size distribution for the original network considering only the encounters that involve oral sex without condom, and mouth kissing (

Returning to our original network, we investigate the effect of varying ρ, and see that the average outbreak size 〈Ω〉 is an approximately linear function of transmission probability (see _{0}) in the original data set. We take the crossing point between the fit of the fraction of infected vertices as a function of ρ to a line and the line of zero secondary infections as an estimate of the threshold value. We see in _{0}, which is our estimated threshold value for this contact pattern. This threshold seems slightly smaller for the RDT, but significantly smaller for the SN and VM models (

Panel A displays the average outbreak size 〈Ω〉 as a function of the transmission probability. The line is a linear trend least square fitted to the data in the interval 0.3≤ρ≤1. The abscissa is in log-scale. Panel B shows the threshold ρ-value (estimated by the crossing of the linear fitting and the zero-size outbreak line) as a function of the beginning of the sampling window.

Outbreak size versus transmission probabilities for (A–C) SI and (D–F) SIR epidemic models. Each panel shows the results for the empirical and for a random network. The abscissa is in log-scale.

We plot the average outbreak size 〈Ω〉 as a function of the duration of the infective stage δ in

Panel A shows the average outbreak size 〈Ω〉 as a function of the duration of the infective stage _{0}.

Now, fixing the infective stage to δ = 91 days, which is roughly 3 months and well above our estimated threshold of δ*, we perform SIR simulations for different transmission probabilities and compare the outbreak sizes by using the original network, the randomized version (RDT), a static (SN), and a dynamic network (VM) (

Now we turn to the results of the SI_{1}I_{2} simulation of HIV spread. We fix the acute infective period at _{1} = 91 days and study some different combinations of estimated transmission probabilities available in the literature for different societies by using lower (ρ_{1} = 0.005 and ρ_{2} = 0.0005) and higher (ρ_{1} = 0.01 and ρ_{2} = 0.001) bounds _{1}I_{2} model on the actual data is below the epidemic threshold. In ^{–5} of the population) for both these contact patterns. For the RDT model and ρ_{1} = ρ_{2} = 0.01, the system is just above the epidemic threshold, as can be seen by its convex curve in _{1} = ρ_{2} = 0.01 and ρ_{1} = 0.01, ρ_{2} = 0.001 curves are almost congruent. For the RDT contact structure, the more homogeneous temporal pattern allows the chronic infection to play a greater role, so these two curves diverge after about 200 days, which is about the average interval between two consecutive contacts.

The increase in the number of infected vertices, Ω, using the simulated SI_{1}_{2}

We simulate the spread of infection in what is probably the largest network of self-reported sexual contacts yet recorded. Our data come from a web community of sex buyers who discuss their encounters with escorts. Although the network is spread out over twelve cities, it is to a large extent connected so that a disease could spread from most parts of the system to most other parts. As with any result based on a subset of a network, we should be cautious about extrapolating our results to the entire society, especially since it is hard to compensate for missing links with the information we have. The escorts in our dataset make up about one percent of all Brazilian sex sellers (of a total of about one million _{critical} = 〈^{2}〉 (where _{critical}∼0.043, using our network. The network structure (apart from the contact-rate distribution), on the other hand, slows down outbreaks. Our network has a high density of short cycles and community structure, reflecting the fact that most sex buyers buy sex in one region, presumably their hometown. Both factors, many short clusters and distinct communities are known to slow down diffusion in networks

Most of our analysis is at a general STI level and indicates that our network is not dense enough to support STI outbreaks for chronic diseases with transmission probabilities lower than ρ = 0.19. The fact that endemic diseases with arguably lower transmission probabilities exist points to the importance of the background sexual contacts. Because of the incompleteness of our data, this does not completely exclude the possibility of escort prostitution as a reservoir of STIs, but it points to a more complex picture. In the support information (_{0} of STIs by a few percent. We also exemplify how temporal structures can affect the spread of a specific pathogen, HIV-1, by simulations of a refined compartmental model. The simulation results indicate that our empirical network alone cannot sustain an outbreak of HIV-1. In general agreement with empirical research

We believe that the study of temporal aspects of contact patterns is, in general, a promising direction for the future. We intend to investigate how far our conclusions can be generalized to other types of cultures, other forms of commercial sex, and hopefully to non-commercial sexual contact patterns.

We provide the full dataset in .csv format containing the sexual network used in this paper. Specific information about the format of the data is inside the file.

(1.20 MB CSV)

Augmenting well-mixed models. We make a short analysis of the contribution of the studied commercial sexual network to epidemics if this network is embedded in a larger network of sexual contacts.

(0.20 MB PDF)