*“It might be informative to look at other units of measurements than the individual. A relational perspective might help to understand how people adjust to the risk of HIV in relation to the different types of and moments in their relationships. This should also be done in relation to other aspects, such as sexually transmitted disease prevention and birth control.”* ^{1}

THE INFECTION RISK of an individual is determined, among others factors, by the individual's position in a network. Therefore, not only the local position of an individual is important, but the overall structure of the network also determines the risk of infection. A well-known example for the effect of network position is monogamous women whose husbands visit commercial sex workers and, in turn, transmit HIV infection to their wives. Here, persons who do not practice risky behavior themselves are at risk of infection because of their location in the network at a short distance from the high-risk core group. Prevention efforts directed toward these persons alone would have little effect. Only prevention that targets the partnerships between those women and their husbands or the contacts of husbands with commercial sex workers would reduce the risk to their wives. Therefore, prevention has to be viewed in a larger context than the behavior of individuals.

Methodologically, however, it has proven to be difficult to analyze the effect of network structure on disease transmission. Although a network can be visualized well as a graph, where the nodes represent individuals of the population and the links represent their sexual partnerships, a mathematical analysis of such a graph structure that could contribute to the understanding of disease transmission is only at the beginning of its development. One of the most important challenges is to find overall measures of network structure that can be measured in the field and that give essential information about the way an infection spreads in the network. From research in social sciences, three main characteristics of a network emerge: (1) the mean and the variance of the number of links per node, (2) the transitivity of the network, and (3) the characteristic path length. Although the first is clearly a description of the local neighborhood of nodes and can be measured in sexual-behavior surveys (the mean and variance of the number of partners), the others are global measures of network structure.

Transitivity describes the probability that two nodes that are linked to the same third node link to each other. In other words, it is the probability that two sex partners of a person also have a sexual relationship with each other. In a heterosexual population, one would have to extend this definition to include cycles of four persons: it is the probability that two male sexual partners of one woman both have sex with the same other woman. In a wider sense of transitivity, one could also consider larger cycles. Transitivity describes network structure in a somewhat wider neighborhood that includes not only one's nearest neighbors, but also neighbors at a given short distance of a person, where distance is defined as the number of links between two nodes. The third potentially important network measure is the characteristic path length in a network, which is defined as the average length of the shortest possible path between any two individuals in the population, given that there is some path connecting them. Clearly, this is a global network measure that gives information about how strongly clustered a network is. As estimation of the average path length necessitates knowledge about possibly long chains of links, it is a quantity that is difficult to measure in the field.

If one wants to use the graph-theory paradigm to describe sexual networks, there are other problems in defining what one means with a given graph structure. One problem is the clear definition of a link between two nodes. When do we say that two individuals have a sexual partnership? They may have sexual encounters at very irregular time intervals. Do they have one partnership or several partnerships after each other? When do those partnerships begin and end? Another question is whether the graph should only describe the network as it exists at a given time, or whether it should include all links that were present over some interval (e.g., during a time interval in the order of the infectious period of a given infection).

As these deliberations indicate, discussing network structure and its effects on disease transmission is fraught with a number of difficulties, among others is the clear definition of local and global network effects, identification of relevant network measures, and the estimation of those measures in real sexual networks. Important tools used to gain insight to those complex structures and interactions are mathematical and simulation models. Although it might be an unattainable goal to describe sexual networks in the complexity as they occur in human populations, modeling can help to obtain insight to the patterns of disease transmission on networks and, therefore, may provide valuable understanding of why some prevention measures work better than others. It is the abstraction and possible simplicity of models that renders them unrealistic, but also provides the power to isolate the essential features of the dynamics of disease transmission in different settings.

The purpose of this report is to introduce some modeling approaches for sexually transmitted diseases (STDs) that include partnership dynamics or larger network structure, and discuss implications from those models for prevention and intervention. After giving a short introduction to the types of models that have been developed in the literature, we discuss the implications of some models that include some form network structure; namely, pair-formation models and stochastic network models. Contact tracing as one of the most important partnership based prevention method is then discussed, and I show how models can be used to investigate the effectiveness of contact tracing within a population-based screening program. Finally, possible future research questions that should be tackled if modeling results are to be used for planning public health policy are reviewed.

**Types of Model Structure**

Traditional modeling approaches to STDs are based on the so-called susceptible, infected (and infectious), or removed (immune or dead) (SIR) model,^{2,3} which describes a population in which individuals can be susceptible, infected, or removed. The population is assumed to mix homogeneously (i.e., each individual has the same probability of contacting each other individual). Contacts are assumed to have no duration, and individuals never contact each other more than once. Obviously, those assumptions are not appropriate for describing contacts within sexual relationships, where populations are usually strongly heterogeneous in their sexual activity and in the choice of partners, and where partnerships can last long, such that many sexual contacts take place between the same two persons.

A common approach in dealing with population heterogeneity is taken in mixing models. In those models, the population is stratified by level of sexual activity. The model then describes the contact rates within and between the population groups with different sexual-activity levels. Models of that kind were first introduced by those who coined the term *core group* for a small but highly sexually active subgroup in a population.^{4} The core-group concept has since been successful in explaining the continuing persistence of STDs on relatively low levels of prevalence in populations where, on average, the rate of partner change is low. A comprehensive introduction to mixing models can be found elsewhere.^{5}

Models that explicitly take partnership duration into account have been developed in mathematical demography.^{6} In the context of infectious diseases, those so-called pair-formation models were first introduced by Dietz and Hadeler.^{7} In addition to being susceptible, infected, or removed, an individual is now also described as being single or paired with another individual in the population. In other words, the model distinguishes between singles and pairs and describes the transition between those states in terms of pair-formation and separation rates (Figure 1). A pair can consist of two susceptible persons, two infected persons, or one susceptible and one infected individual. Disease transmission only takes place within the latter type of pairs, and on transmission such a pair moves into the class of pairs with two infected individuals. Including the pair-formation mechanism in a model has several effects on the spread of an infection. One such effect is that susceptible persons who are in a long-lasting partnership with another susceptible individual are effectively protected from infection. The other effect is that infected individuals paired with another infected individual “waste” contacts without infecting others. Finally, the average duration spent as a single influences the risk of being infected or transmitting the infection to others. In other words, there is a close relationship between those parameters describing the partnership dynamics, such as the partnership duration and the average time between two partnerships, and the expected course of an epidemic or the endemic prevalence. This can be expressed in terms of the basic reproduction ratio, R_{0}, and its dependence on pair-formation and separation rates.

When taking partnership duration into account, one is immediately confronted with the question of possible overlap of partnerships, something that is irrelevant if partnerships are assumed to be infinitely short. Pair-formation models in the strict sense are based on the assumption that individuals can have at most one partnership at a time (i.e., the population consists of singles and pairs only). If one wants to relax that condition in the deterministic setting, even in the simplest cases the number of equations and parameters increases rapidly^{8} and models become untractable. There are some first investigations into other analytic ways to solve the problem,^{9,10} but those approaches are not sufficiently developed to provide a modeling framework that is amenable for more realistic settings as they arise in public health applications. Therefore, the only current possibility is to work with simulation models that describe the structure of the sexual partnership network and the transmission of infectious disease through that network.

This approach was volunteered by Blanchard et al,^{11} and has since been developed by Kretzschmar et al.^{12} The basic idea of those models is that a partnership network can be described by a graph: individuals are the nodes of the graph, and their partnerships the links connecting those nodes. Network structure can then be described in terminology of graph theory, using concepts such as the degree distribution, connectivity, and centrality. Unfortunately, only few analytical tools are available for analyzing such models. Therefore, the method of investigation is mainly Monte Carlo simulation of stochastic and individual-based models. This has the advantage of producing output that can directly be compared with survey and epidemiologic data. The disadvantage is the time-consuming necessity to perform a large number of simulation runs and analyzing the results statistically.

**Pair-Formation Models**

Investigating the dynamics of pair-formation models gives insight into the effects of partnership duration on the spread of an infectious disease. When comparing a model without pair formation with a model that includes pair formation for a set of parameters that are realistic for HIV, Dietz^{13} concluded that in a situation with pair formation, the rise of prevalence toward the endemic equilibrium is much slower and the eventual endemic prevalence lower than estimated in a model that ignores partnership duration. Similar results were obtained with a model that included both long-term partnerships and incidental contacts with a small group of commercial sex workers.^{14} In the latter investigation, the endemic prevalence of HIV infection in the model without pairs was approximately twice the prevalence estimated for a model that considers the duration of steady partnerships. However, in those comparisons the value of R_{0} differed for the models with and without partnership duration. The following comparison was made with slightly different results based on the assumption that R_{0} is the same for both situations.

For an infection like HIV, in which infectivity varies greatly during the course of an infection, the position of the infectious person in the network is especially important at the time of high infectiousness. As that is the period of approximately 2 months after infection has taken place, the duration of partnerships and their exclusiveness will be crucial in determining the spread of HIV. For a heterosexual population with mainly monogamous partnerships, the duration of those partnerships will determine how fast HIV can spread in the population. To investigate the relationship between partnership duration and variable infectivity, Kretzschmar and Dietz^{15} investigated a pair-formation model for an infection with a two-stage disease process. To analyze the effects of partnership duration, the model was compared to a model without pair formation (i.e., a model in which contacts were assumed to be instantaneous). For the model with pair formation, one can analyze how the basic reproduction ratio R_{0} depends on the duration of partnerships (Figure 2). If partnerships tend to last very long, R_{0} is less than 1 because while an infected person is tied up in a partnership, she or he cannot infect any other person. Only after separation of the partnership and formation of a new partnership can a new infection be produced. Therefore, the duration of partnerships puts a bound on the number of secundary cases that an infected individual can cause during his or her infectious period. There may also be situations in which very short partnership duration leads to a value of R_{0} of more than 1. In this case, the partnerships do not last long enough to ensure transmission to the partner. It is then the combination of short partnerships with long interpartnership periods during which no transmission is possible that limit the number of secundary cases per primary case.

Based on the assumption that the basic reproduction ratio R_{0} should be independent of the model used, one can compare the exponential growth rate and the endemic equilibrium for a model with partnership dynamics and a model with instantaneous contacts.^{15} For the case of constant infectivity, it can be shown analytically that the exponential growth rate is smaller and, therefore, the doubling time of the epidemic is longer in a model with partnership dynamics than in a model without partnership dynamics. One also observes that for a given R_{0} there can be two values of the doubling time (or the exponential growth rate), depending on the underlying partnership dynamics. However, in the model with partnership dynamics, the endemic prevalence will eventually be higher than that predicted by a model without partnership dynamics. Similarly, for a given value of R_{0} there may be two endemic levels of prevalence, again depending on the underlying partnership dynamics-in this case the partnership duration. This has consequences for the estimation of R_{0} from endemic prevalences; to get a valid estimate, one needs to have information about the partnership dynamics.

Looking specifically at the models including variable infectivity, it is interesting to compare the transient behavior of the models, again under the assumption that they have the same value of R_{0}. If partnership duration is short in the pair-formation model, both models behave similarly. As partnership duration is increased, the behaviors of the models diverge. Already, for a partnership duration of approximately 40 days in the model with partnership dynamics, the level of prevalence only reaches detectable levels after more than 10 years, whereas a model without partnership dynamics predicts an endemic prevalence of more than 30% after 5 years. This example underlines the importance of taking partnership duration into account when modeling the spread of HIV in heterosexual populations in Western Europe and North America.

An important aspect of partnership duration for intervention might be its impact on vaccination in which infectivity in the first phase of infection can be reduced by reduction of viral load. The effectiveness of such a vaccination on the further spread of HIV might depend strongly on the contact patterns in the population to be vaccinated. For example, if most partnerships in this population are longer-lasting monogamous partnerships, then reducing the infectivity in the first phase of infection might change R_{0} only slightly. A similar argument can be made for a situation where safe sex is practiced in the majority of casual partnerships but not in longer-lasting partnerships. A large part of all transmissions would then occur in longer-lasting partnerships, and one would expect similar effects of partnership duration.

**Network Models and Concurrent Partnerships**

The drawback of pair-formation models is their assumption that persons do not have more than one partnership at a time. Although this assumption can be relaxed to some extent,^{8} pair-formation models are not applicable to situations where concurrent partnerships are common. In a population where many concurrent partnerships occur, chains of partnerships are formed and larger network components can exist. Until recently, satisfactory deterministic models for such situations have not been developed. The best approach to date is the stochastic modeling of individual-based network models; individuals and their partnerships are described explicitly and can be depicted as nodes and links of a graph.

A stochastic individual-based model for networks of partnerships has been developed and investigated.^{12,16,17} One of the aims of developing the model was to investigate the importance of concurrent partnerships for the spread of an infectious disease such as HIV through a network. For this investigation, it was necessary to develop methods for measuring the level of concurrency in a population. Concurrency on the individual level is readily defined: two partnerships are concurrent if there is a person that participates in both of them. In graph-theoretical terms, two links of the graph are concurrent if they have a node in common. To lift this definition to the population level, it is convenient to consider the line graph *H* of a graph *G.* If v_{1}, …, v_{n} are the nodes of *G,* and e_{1}, …, e_{m} the links of *G,* then the line graph *H* is defined as follows: the nodes of *H* are the the links of *G,* and two nodes in *H* are connected if the corresponding links in *G* have a node in common. Let |F| be the number of nodes of *H,* and |W| the number of links of *H.* Then, we define the index of concurrency κ as κ = |F|/|W|. In other words, we measure concurrency on the population level by the mean number of links per node (in technical language, the mean degree) of the line graph of the contact graph. This measure can be interpreted as describing the average number of concurrent partnerships per partnership in the contact graph *G.* In the first instance, the definition for κ sounds impractical; however, it can be shown that the following equation holds for κ^{17}: EQUATION (1) where *v* denotes the variance and *m* the mean of the degree distribution (the distribution of number of nodes per link) of *G.* Therefore, it is sufficient to know the distribution of the number of current partners to determine the level of concurrency in the population at that time.

In simulation experiments where the average number of partnerships in the population was kept constant and the distribution of partnerships in the population varied, the impact of concurrent partnerships on disease dynamics could be investigated. The main results were that (1) the exponential growth rate in the beginning phase of the epidemic increased approximately linearly with concurrency index κ; (2) the number of infected individuals at a given time *t* after the beginning of the epidemic increased approximately exponentially with increasing κ; and (3) the number of nodes in the largest network component increased approximately linearly with κ. The latter result leads to an interpretation of the first two results: The increasing connectivity of the network, characterized by the size of the largest network component, decreases the average time between infection of an individual and transmission to another of his or her partners. The limit on the number of secundary cases per index case, which is set by the partnership duration, is circumvented if more partners are available at a time to transmit the infection to. Time between partnerships and partnership duration lose in importance for the dynamics of the infection.

The more widespread existence of long-lasting concurrent partnerships in sub-Saharan Africa might be one possible reason for the fast spread of HIV in heterosexual populations in those regions.^{18,19} This has been investigated using the previous simulation model on the basis of sexual survey data obtained in a field study in Uganda.^{20} The results suggest that during the first 5 years of the epidemic, the level of concurrency as currently observed could have increased HIV prevalence by 26% compared with a population with sequential monogamy. Because there probably has been a behavior change toward fewer partners and fewer concurrent partnerships, the impact of these two factors on the fast rise of HIV prevalence in heterosexual populations in sub-Saharan Africa was probably much stronger. This is shown in Figure 3 (top), where the impact of a larger number of partners and more concurrent partnerships on the number of persons infected after 5 years of transmission was explored by simulation. The results show that an increase in the number of partners without increasing the level of concurrency has little effect on the number of persons infected, whereas increasing concurrency without changing the average number of partners has a noticeable effect. When both factors are increased simultaneously, the increasing number of partners amplifies the effect of increasing concurrency. The reason for the faster transmission of infection lies in the changes of network structure induced by the concurrent partnerships (Figure 3 bottom); with increasing concurrency the size of the largest component increases, indicating the increased clustering of individuals and, therefore, shorter transmission routes.

When designing prevention programs that aim to change sexual behavior, not only the number of partners should be decreased, but the overlap of partnerships should also be discouraged. Equivalently, when promoting condom use, encouraging people to use condoms with all partners outside of their steady partnership becomes even more important. These results also indicate that aiming to break up larger network components is an effective way of impeding further transmission. This has been discussed by Rothenberg et al,^{21} who suggest that segmentation of social networks may be an important method of interrupting transmission. One could think of programs that aim to split larger network components into smaller subunits. This can be reached without changing the frequency of risk behavior or the average number of partners per person. For example, by closing shooting galleries for intravenous drug users, the large network of users who frequented this location might be split into smaller groups that share among each other. Such measures reduce the connectivity of the network and slow the spread of the infection (HIV). Similar mechanisms might play a role in core groups of highly sexually active persons.

**Contact Tracing in Networks**

One type of STD intervention that clearly is directed toward partnerships and that takes network structure into account is contact tracing (i.e., notification and treatment of partners of infected persons). Surprisingly, little theoretical research has been done to evaluate the impact of contact tracing on decreasing incidence and prevalence of STDs or other infectious diseases. In retrospect, this finding is not surprising, because modeling contact tracing requires one to keep track of partnerships, which means that one needs a model that describes partnership dynamics or partnership networks. The type of model that is needed depends on how contact tracing is implemented. If current partners of infected individuals are to be notified, a model that includes partnership duration is needed. If past partnerships are included, a model that keeps track of the contact network over a given interval is needed.

The effectiveness of contact tracing can be understood in terms of network structure: it explicitly uses the position of the infected index case in the contact network. The infected index case might be a member of a highly active core group. In that case, the infected person has had many partners, all of whom could be traced and may be infected. If the infected index case is not a member of the core group, chances are that his or her partners are members of the core group, or at least that the chain of contacts leading to the core group is short. In other words, tracing this person's partners possibly leads to infected individuals who are transmitting to many others. Also, contact tracing reduces the probability of reinfection of the treated index case. In that sense, contact tracing can be interpreted as treatment of infected partnerships instead of treatment of individuals.

Contact tracing can be implemented if a symptomatically infected person seeks health care and is treated, but also in combination with systematic screening programs that are designed to find asymptomatically infected persons (e.g., *Chlamydia trachomatis,* gonorrhea). Recently, a stochastic network simulation model was used to investigate screening programs in combination with contact tracing based on a pilot study conducted in Amsterdam.^{22,23} In the model and in the pilot study, a systematic screening was implemented for the general population to find asymptomatic infections of *C trachomatis.* If a person was found to be infected, an effort was made to contact his or her partner(s) and to offer treatment to them. Using the model various variants of the screening programs were investigated and compared: screening of different age groups; screening of only women, only men, or both; and different success rates for contact tracing. The sexual network in the model population included a highly sexually active core group, two types of partnerships (short-term casual, long-lasting steady partnerships), and concurrent partnerships of core group members.

Contact tracing was found to contribute significantly to the success of the screening program. This is true when women and men in the 15-year to 24-year age group are screened, where one might think that partners are found by the screening program. When partner referral rates of 44% were assumed for men (i.e., 44% of all female partners of men found by screening are referred and receive treatment) and 65% for women, the prevalence of asymptomatic *C trachomatis* infections in women could be reduced from 4.2% to 1.4% in 10 years; without partner referral, the reduction was only 2.2%. In the former scenario, 28% of all asymptomatically infected persons were found by contact tracing. Prevalence could be reduced by more than 70% compared with the prescreening value if contact tracing could be made even more effective (partner referral rate of 97.5% for women, 66% for men) (Figure 4).

As a measure for the additional effect of partner referral, I computed the ratio of prevalence reduction with partner referral to prevalence reduction without partner referral. If *P* _{pre} is the prescreening prevalence and *P* _{x} is the postscreening prevalence (after 10 years of screening) with *x*% partner referral, and *P* _{0} the postscreening prevalence without partner referral, then EQUATION (2). For the reduction of the prevalence of asymptomatic infections in women and for the values of *x* (44% for men, 65% for women), if both men and women are screened, E_{p} = 1.41; if only women are screened, E_{p} = 1.44; and if only men are screened, E_{p} = 2.14. This finding shows that including partner referral can make a screening program almost 50% more effective in reducing prevalence if women are included in the screening process. If only men are screened, the additional effect of partner referral is more than 100%. If referral rates are lower (22% for men, 32.5% for women), the additional effect of partner referral is still approximately 20% (E_{p} = 1.18 for screening men and women, E_{p} = 1.21 for screening only women). The reasons for this high impact of contact tracing are twofold. First, by contact tracing, partners can be found in age groups that are not included in the screening program, and there is a good chance that those individuals are found that contribute most to transmission. Second, reinfection of the index case is prevented, and in this way the transmission chain is effectively interrupted. Contact tracing also made a significant contribution when men and women were included in the screening program, because the screening as implemented here relied on people visiting the general practitioner for other reasons. Therefore, the fraction of the population reached depends on the frequency of visiting a general practitioner, which is considerably lower for men than for women, especially in the younger age groups.

This report has only been a first investigation into the effects of contact tracing. The tracing considered in the previous model is simple: it only includes current and direct partners, and does not trace longer chains of infected persons. In a more theoretical study,^{24} the effectiveness of contact tracing is investigated for a stochastic and individual-based SIR model. The model is kept simple enough to allow for derivation of some analytical results. A striking result for the special case that there is no spontaneous recovery from infection is the following: if p_{c} denotes the fraction of infected partners that are found by contact tracing per index case, to bring the basic reproduction number below 1 it is necessary that EQUATION (3), a formula that is analogous to the formula determining the critical vaccination coverage needed to eliminate an infection by universal vaccination. Although both vaccination and contact tracing influence the reproduction ratio of an infectious disease, they do so in different ways. Vaccination reduces the fraction of the population that is susceptible, thereby reducing the number of effective contacts in which disease transmission can take place. Contact tracing reduces the average duration of the infectious period because persons found by contact tracing will be treated and lose infectiousness. It is not clear how this formula can be extended to more general situations. If there is spontaneous recovery, then transmission chains cannot be followed by contact tracing of those recovered individuals, which makes contact tracing less effective, possibly so that even with p_{c} = 1, the basic reproduction ratio R_{0} cannot be brought below 1.

The model used to obtain those results is simple and does not include any population heterogeneity or complex network structure. Also, it has not yet been adapted to describe a specific disease with its characteristics in transmission and course of infection. However, the model does lead to some insight into why and under which circumstances contact tracing is an important addition to other public health measures.

A practical problem regarding the use of the parameter p_{c} as a measure of the efficacy of contact tracing is that, in general, the total number of infected contacts is not known; therefore, the estimation of which fraction of those contacts can be reached by contact tracing is not possible. Given a specific infectious disease, it might be possible to design studies to get an idea of the distribution of the number of infected contacts. For example, in the case of tuberculosis, information obtained from the analysis of DNA fingerprinting has been used to estimate how many secundary infections are actually found by contact tracing.^{25,26} For tuberculosis, those estimates have been low (5-10%). For STDs for which contacts are more clearly defined, one might hope to obtain a higher efficacy of contact tracing.

**Conclusions and Future Research Needs**

The main conclusion from the above considerations is that it is necessary to take partnership duration and the structure of the partnership network into account when investigating the efficacy of STD prevention and intervention. Neglecting those aspects of STD-transmission dynamics may lead to false expectations about the impact of a prevention and intervention program. Furthermore, one can conclude that designing prevention and intervention programs that target, besides the individual at risk of being infected, also his or her partners, or larger network components, may significantly improve the outcome of those programs. One could think of prevention programs being designed on three levels: (1) the level of individuals by targeting a person's risk-taking behavior; (2) on the level of partnerships by targeting how two individuals interact with each other (e.g., condom use); and (3) the network level by targeting groups of individuals and their clustering (e.g., by providing small-scale meeting places and supporting local friendship networks).

An important question in estimating the possible impact of such prevention measures on prevalence and incidence is the interaction of a specific disease with a given network structure: some infections might spread faster on certain types of networks than others. An example is the influence of partnership duration on the transmission of an infection with variable infectivity, such as HIV infection. If the main fraction of infectivity is concentrated in a short time interval after infection, it is crucial for further transmission whether individuals tend to change their partners or have concurrent partnerships in that period. More research is needed for other STDs to unravel the capacity of specific diseases to spread in networks with specific structural properties.

Furthermore, the network approach has not yet really been used for developing new prevention strategies.^{27} The reasons are twofold. First, we simply do not yet have a good understanding of the relationship between network structure and disease transmission. Second, targeting larger network structures, such as connected components, is more difficult than targeting individuals and necessitates a deeper knowledge of global network structure. Hopefully, future insights into the relationship of network structure and the spread of STDs will lead to the development of new and effective prevention strategies.

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