A moment closure model for sexually transmitted disease transmission through a concurrent partnership network

Concurrency measures in the era of temporal network epidemiology: a review

Diseases spread over temporal networks of interaction events between individuals. Structures of these temporal networks hold the keys to understanding epidemic propagation. One early concept of the literature to aid in discussing these structures is concurrency-quantifying individuals' tendency to form time-overlapping 'partnerships'. Although conflicting evaluations and an overabundance of operational definitions have marred the history of concurrency, it remains important, especially in the area of sexually transmitted infections. Today, much of theoretical epidemiology uses more direct models of contact patterns, and there is an emerging body of literature trying to connect methods to the concurrency literature. In this review, we will cover the development of the concept of concurrency and these new approaches.

Partnership dynamics in mathematical models and implications for representation of sexually transmitted infections: a review

Mathematical models of sexually transmitted disease (STI) are increasingly relied on to inform policy, practice, and resource allocation. Because STI transmission requires sexual contact between two or more people, a model's ability to represent the dynamics of sexual partnerships can influence the validity of findings. This ability is to a large extent constrained by the model type, as different modeling frameworks vary in their capability to capture patterns of sexual contact at individual, partnership, and network levels. In this paper, we classify models into three groups: compartmental, individual-based, and statistical network models. For each framework, we describe the basic model structure and discuss key aspects of sexual partnership dynamics: how and with whom partnerships are formed, partnership duration and dissolution, and temporal overlap in partnerships (concurrency). We illustrate the potential implications of accurately accounting for partnership dynamics, but these effects depend on characteristics of both the population and pathogen; the combined impact of these partnership and epidemiologic dynamics can be difficult to predict. While each of the reviewed model frameworks may be appropriate to inform certain research or policy questions, modelers and consumers of models should carefully consider the implications of sexual partnership dynamics for the questions under study.

Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptible individuals and infected individuals are distinguished from the population of three types of pairs of individuals: both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. We use perturbation expansion to formally derive uniformly valid approximations for the dynamics of the total infected and susceptible populations under different conditions including combinations of fast association, fast transmission, and fast dissociation limits. The effective equations are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms, such as those used in frequency-dependent models. Our results represent submodels that show how effective nonlinear transmission can arise from pairing dynamics and are juxtaposed with density-based mass-action and frequency-based models.

The Impact of Pre-exposure Prophylaxis for Human Immunodeficiency Virus on Gonorrhea Prevalence

Pre-exposure prophylaxis (PrEP) has been shown to be highly effective in reducing the risk of HIV infection in gay and bisexual men who have sex with men (GbMSM). However, PrEP does not protect against other sexually transmitted infections (STIs). In some populations, PrEP has also led to riskier behavior such as reduced condom usage, with the result that the prevalence of bacterial STIs like gonorrhea has increased. Here, we develop a compartmental model of the transmission of HIV and gonorrhea and the impacts of PrEP, condom usage, STI testing frequency and potential changes in sexual risk behavior stemming from the introduction of PrEP in a population of GbMSM. We find that introducing PrEP causes an increase in gonorrhea prevalence for a wide range of parameter values, including at the currently recommended frequency of STI testing once every three months for individuals on PrEP. Moreover, the model predicts that a higher STI testing frequency alone is not enough to prevent a rise in gonorrhea prevalence, unless the testing frequency is increased to impractical levels. However, testing every 2 months in combination with a 10-25 % reduction in risky behavior by individuals on PrEP would maintain gonorrhea prevalence at pre-PrEP levels. The results emphasize that programs making PrEP more available should be accompanied by efforts to support condom usage and frequent STI testing, in order to avoid an increase in the prevalence of gonorrhea and other bacterial STIs.

Improving pairwise approximations for network models with susceptible-infected-susceptible dynamics

Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.

Exact and approximate formulas for contact tracing on random trees

We consider a stochastic susceptible-infected-recovered (SIR) model with contact tracing on random trees and on the configuration model. On a rooted tree, where initially all individuals are susceptible apart from the root which is infected, we are able to find exact formulas for the distribution of the infectious period. Thereto, we show how to extend the existing theory for contact tracing in homogeneously mixing populations to trees. Based on these formulas, we discuss the influence of randomness in the tree and the basic reproduction number. We find the well known results for the homogeneously mixing case as a limit of the present model (tree-shaped contact graph). Furthermore, we develop approximate mean field equations for the dynamics on trees, and - using the message passing method - also for the configuration model. The interpretation and implications of the results are discussed.

Concurrency of partnerships, consistency with data, and control of sexually transmitted infections

Sexually transmitted infections (STIs) are a globally increasing public health problem. Mathematical models, carefully matched to available epidemiological and behavioural data, have an important role to play in predicting the action of control measures. Here, we explore the effect of concurrent sexual partnerships on the control of a generic STI with susceptible-infected-susceptible dynamics. Concurrency refers to being in more than one sexual partnership at the same time, and is difficult to measure accurately. We assess the impact of concurrency through the development of three nested pair-formation models: one where infection can only be transmitted via stable sexual partnerships, one where infection can also be transmitted via casual partnerships between single individuals, and one where those individuals in stable partnerships can also acquire infection from casual partnerships. For each model, we include the action of vaccination before sexual debut to inform about the ability to control. As expected, for a fixed transmission rate, concurrency increases both the endemic prevalence of infection and critical level of vaccination required to eliminate the disease significantly. However, when the transmission rate is scaled to maintain a fixed endemic prevalence across models, concurrency has a far smaller impact upon the critical level of vaccination required. Further, when we also constrain the models to have a fixed number of new partnerships over time (both long-term and casual), then increasing concurrency can slightly decrease the critical level of vaccination. These results highlight that accurate measures and models of concurrency may not always be needed for reliable forecasts when models are closely matched to prevalence data. We find that, while increases in concurrency within a population are likely to generate public-health problems, the inclusion of concurrency may be unnecessary when constructing models to determine the efficacy of the control of STIs by vaccination.

Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions

Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number R0 and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

Concurrency-Induced Transitions in Epidemic Dynamics on Temporal Networks

Social contact networks underlying epidemic processes in humans and animals are highly dynamic. The spreading of infections on such temporal networks can differ dramatically from spreading on static networks. We theoretically investigate the effects of concurrency, the number of neighbors that a node has at a given time point, on the epidemic threshold in the stochastic susceptible-infected-susceptible dynamics on temporal network models. We show that network dynamics can suppress epidemics (i.e., yield a higher epidemic threshold) when the node's concurrency is low, but can also enhance epidemics when the concurrency is high. We analytically determine different phases of this concurrency-induced transition, and confirm our results with numerical simulations.

Pair formation models for sexually transmitted infections: A primer

For modelling sexually transmitted infections, duration of partnerships can strongly influence the transmission dynamics of the infection. If partnerships are monogamous, pairs of susceptible individuals are protected from becoming infected, while pairs of infected individuals delay onward transmission of the infection as long as they persist. In addition, for curable infections re-infection from an infected partner may occur. Furthermore, interventions based on contact tracing rely on the possibility of identifying and treating partners of infected individuals. To reflect these features in a mathematical model, pair formation models were introduced to mathematical epidemiology in the 1980's. They have since been developed into a widely used tool in modelling sexually transmitted infections and the impact of interventions. Here we give a basic introduction to the concepts of pair formation models for a susceptible-infected-susceptible (SIS) epidemic. We review some results and applications of pair formation models mainly in the context of chlamydia infection.

Forward reachable sets: Analytically derived properties of connected components for dynamic networks

Formal analysis of the emergent structural properties of dynamic networks is largely uncharted territory. We focus here on the properties of forward reachable sets (FRS) as a function of the underlying degree distribution and edge duration. FRS are defined as the set of nodes that can be reached from an initial seed via a path of temporally ordered edges; a natural extension of connected component measures to dynamic networks. Working in a stochastic framework, we derive closed-form expressions for the mean and variance of the exponential growth rate of the FRS for temporal networks with both edge and node dynamics. For networks with node dynamics, we calculate thresholds for the growth of the FRS. The effects of finite population size are explored via simulation and approximation. We examine how these properties vary by edge duration and different cross-sectional degree distributions that characterize a range of scientifically interesting normative outcomes (Poisson and Bernoulli). The size of the forward reachable set gives an upper bound for the epidemic size in disease transmission network models, relating this work to epidemic modeling (Ferguson, 2000; Eames, 2004).

Phylodynamics on local sexual contact networks

Phylodynamic models are widely used in infectious disease epidemiology to infer the dynamics and structure of pathogen populations. However, these models generally assume that individual hosts contact one another at random, ignoring the fact that many pathogens spread through highly structured contact networks. We present a new framework for phylodynamics on local contact networks based on pairwise epidemiological models that track the status of pairs of nodes in the network rather than just individuals. Shifting our focus from individuals to pairs leads naturally to coalescent models that describe how lineages move through networks and the rate at which lineages coalesce. These pairwise coalescent models not only consider how network structure directly shapes pathogen phylogenies, but also how the relationship between phylogenies and contact networks changes depending on epidemic dynamics and the fraction of infected hosts sampled. By considering pathogen phylogenies in a probabilistic framework, these coalescent models can also be used to estimate the statistical properties of contact networks directly from phylogenies using likelihood-based inference. We use this framework to explore how much information phylogenies retain about the underlying structure of contact networks and to infer the structure of a sexual contact network underlying a large HIV-1 sub-epidemic in Switzerland.

Phylodynamic models relate the branching pattern of a pathogen’s phylogenetic tree to the tree-like growth of an epidemic as it spreads through a host population. Such models are increasingly used to learn about the epidemiology of different pathogens. We extend current models to consider the structure of host contact networks—the web of physical interactions through which pathogens spread. By considering how local interactions among hosts shape the phylogeny of a pathogen, our models offer a “pathogen’s eye view” of these networks. Our models also provide a statistical framework that can be used to infer network structure directly from phylogenies, which we use to estimate the properties of a sexual contact network in Switzerland from a HIV phylogeny.

A Comparison of Two Mathematical Modeling Frameworks for Evaluating Sexually Transmitted Infection Epidemiology

BACKGROUND

Different models of sexually transmitted infections (STIs) can yield substantially different conclusions about STI epidemiology, and it is important to understand how and why models differ. Frequency-dependent models make the simplifying assumption that STI incidence is proportional to STI prevalence in the population, whereas network models calculate STI incidence more realistically by classifying individuals according to their partners' STI status.

METHODS

We assessed a deterministic frequency-dependent model approximation to a microsimulation network model of STIs in South Africa. Sexual behavior and demographic parameters were identical in the 2 models. Six STIs were simulated using each model: HIV, herpes, syphilis, gonorrhea, chlamydia, and trichomoniasis.

RESULTS

For all 6 STIs, the frequency-dependent model estimated a higher STI prevalence than the network model, with the difference between the 2 models being relatively large for the curable STIs. When the 2 models were fitted to the same STI prevalence data, the best-fitting parameters differed substantially between models, with the frequency-dependent model suggesting more immunity and lower transmission probabilities. The fitted frequency-dependent model estimated that the effects of a hypothetical elimination of concurrent partnerships and a reduction in commercial sex were both smaller than estimated by the fitted network model, whereas the latter model estimated a smaller impact of a reduction in unprotected sex in spousal relationships.

CONCLUSIONS

The frequency-dependent assumption is problematic when modeling short-term STIs. Frequency-dependent models tend to underestimate the importance of high-risk groups in sustaining STI epidemics, while overestimating the importance of long-term partnerships and low-risk groups.

Disease dynamics and costly punishment can foster socially imposed monogamy

Socially imposed monogamy in humans is an evolutionary puzzle because it requires costly punishment by those who impose the norm. Moreover, most societies were—and are—polygynous; yet many larger human societies transitioned from polygyny to socially imposed monogamy beginning with the advent of agriculture and larger residential groups. We use a simulation model to explore how interactions between group size, sexually transmitted infection (STI) dynamics and social norms can explain the timing and emergence of socially imposed monogamy. Polygyny dominates when groups are too small to sustain STIs. However, in larger groups, STIs become endemic (especially in concurrent polygynist networks) and have an impact on fertility, thereby mediating multilevel selection. Punishment of polygynists improves monogamist fitness within groups by reducing their STI exposure, and between groups by enabling punishing monogamist groups to outcompete polygynists. This suggests pathways for the emergence of socially imposed monogamy, and enriches our understanding of costly punishment evolution.

Many human societies transitioned from polygyny to socially imposed monogamy as group sizes increased. Using a simulation model, the authors show that sexually transmitted infections impose heavier fitness penalties on polygynists as group size grows, enabling monogamists who punish polygyny to thrive in large groups.

Coupled disease-behavior dynamics on complex networks: A review

It is increasingly recognized that a key component of successful infection control efforts is understanding the complex, two-way interaction between disease dynamics and human behavioral and social dynamics. Human behavior such as contact precautions and social distancing clearly influence disease prevalence, but disease prevalence can in turn alter human behavior, forming a coupled, nonlinear system. Moreover, in many cases, the spatial structure of the population cannot be ignored, such that social and behavioral processes and/or transmission of infection must be represented with complex networks. Research on studying coupled disease-behavior dynamics in complex networks in particular is growing rapidly, and frequently makes use of analysis methods and concepts from statistical physics. Here, we review some of the growing literature in this area. We contrast network-based approaches to homogeneous-mixing approaches, point out how their predictions differ, and describe the rich and often surprising behavior of disease-behavior dynamics on complex networks, and compare them to processes in statistical physics. We discuss how these models can capture the dynamics that characterize many real-world scenarios, thereby suggesting ways that policy makers can better design effective prevention strategies. We also describe the growing sources of digital data that are facilitating research in this area. Finally, we suggest pitfalls which might be faced by researchers in the field, and we suggest several ways in which the field could move forward in the coming years.

Sexual behavior, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways

There has been growing use of highly active antiretroviral treatment (HAART) for HIV and significant progress in developing prophylactic HIV vaccines. The simplest theories of counterproductive behavioral responses to such interventions tend to focus on single feedback mechanisms: for instance, HAART optimism makes infection less scary and thus promotes risky sexual behavior. Here, we develop an agent based, age-structured model of HIV transmission, risk perception, and partner selection in a core group to explore behavioral responses to interventions. We find that interventions can activate not one, but several feedback mechanisms that could potentially influence decision-making and HIV prevalence. In the model, HAART increases the attractiveness of unprotected sex, but it also increases perceived risk of infection and, on longer timescales, causes demographic impacts that partially counteract HAART optimism. Both HAART and vaccination usually lead to lower rates of unprotected sex on the whole, but intervention effectiveness depends strongly on whether individuals over- or under-estimate intervention coverage. Age-specific effects cause sexual behavior and HIV prevalence to change in opposite ways in old and young age groups. For complex infections like HIV-where interventions influence transmission, demography, sexual behavior and risk perception-we conclude that evaluations of behavioral responses should consider multiple feedback mechanisms.

An Epidemic Patchy Model with Entry-Exit Screening

A multi-patch SEIQR epidemic model is formulated to investigate the long-term impact of entry-exit screening measures on the spread and control of infectious diseases. A threshold dynamics determined by the basic reproduction number R₀ is established: The disease can be eradicated if R₀ < 1, while the disease persists if R₀ > 1. As an application, six different screening strategies are explored to examine the impacts of screening on the control of the 2009 influenza A (H1N1) pandemic. We find that it is crucial to screen travelers from and to high-risk patches, and it is not necessary to implement screening in all connected patches, and both the dispersal rates and the successful detection rate of screening play an important role on determining an effective and practical screening strategy.

Romantic Relationship Dynamics of Urban African American Adolescents: Patterns of Monogamy, Commitment, and Trust

Relationship dynamics develop early in life and are influenced by social environments. STI/HIV prevention programs need to consider romantic relationship dynamics that contribute to sexual health. The aim of this study was to examine monogamous patterns, commitment, and trust in African American adolescent romantic relationships. The authors also focused on the differences in these dynamics between and within gender. The way that such dynamics interplay in romantic relationships has the potential to influence STI/HIV acquisition risk. In-depth interviews were conducted with 28 African American adolescents aged 14 to 21 living in San Francisco. Our results discuss data related to monogamous behaviors, expectations, and values; trust and respect in romantic relationships; commitment to romantic relationships; and outcomes of mismatched relationship expectations. Incorporating gender-specific romantic relationships dynamics can enhance the effectiveness of prevention programs.

HIV transmissions by stage and sex role in long-term concurrent sexual partnerships

Most mathematical models used to examine the role of different stages of human immunodeficiency virus (HIV) infection unrealistically assume that HIV is transmitted through one-off contacts or that transmission rates are the same between males and females. We sought to examine whether inferences from previous models are robust to the relaxation of those unrealistic assumptions. We developed a model of HIV transmissions through sexual partnerships assuming that (1) sexual partnerships have variable duration, (2) sexual partnerships are concurrent, and (3) the male-to-female transmission rate is higher than the female-to-male transmission rate, with a focus on the third assumption. Assuming a higher rate for male-to-female than female-to-male transmissions decreases the overall transmission of HIV but increases the equilibrium fraction of transmissions during primary HIV infection (PHI) in long-term partnerships, compared to the case where transmission rates are assumed to be symmetric between males an females. Previous modeling studies that assume symmetric transmission rates between males and females may have overestimated the overall spread of HIV, but underestimated the relative contribution of PHI. To make robust inferences on the role of different stages of HIV infection in the sexual spread of HIV, models should take into account that transmission rates may be asymmetric by sex.

Dynamics and control of foot-and-mouth disease in endemic countries: a pair approximation model

Previous mathematical models of spatial farm-to-farm transmission of foot and mouth disease (FMD) have explored the impacts of control measures such as culling and vaccination during a single outbreak in a country normally free of FMD. As a result, these models do not include factors that are relevant to countries where FMD is endemic in some regions, like long-term waning natural and vaccine immunity, use of prophylactic vaccination and disease re-importations. These factors may have implications for disease dynamics and control, yet few models have been developed for FMD-endemic settings. Here we develop and study an SEIRV (susceptible-exposed-infectious-recovered-vaccinated) pair approximation model of FMD. We focus on long term dynamics by exploring characteristics of repeated outbreaks of FMD and their dependence on disease re-importation, loss of natural immunity, and vaccine waning. We find that the effectiveness of ring and prophylactic vaccination strongly depends on duration of natural immunity, rate of vaccine waning, and disease re-introduction rate. However, the number and magnitude of FMD outbreaks are generally more sensitive to the duration of natural immunity than the duration of vaccine immunity. If loss of natural immunity and/or vaccine waning happen rapidly, then multiple epidemic outbreaks result, making it difficult to eliminate the disease. Prophylactic vaccination is more effective than ring vaccination, at the same per capita vaccination rate. Finally, more frequent disease re-importation causes a higher cumulative number of infections, although a lower average epidemic peak. Our analysis demonstrates significant differences between dynamics in FMD-free settings versus FMD-endemic settings, and that dynamics in FMD-endemic settings can vary widely depending on factors such as the duration of natural and vaccine immunity and the rate of disease re-importations. We conclude that more mathematical models tailored to FMD-endemic countries should be developed that include these factors.

Individual and population level effects of partner notification for Chlamydia trachomatis

Partner notification (PN or contact tracing) is an important aspect of treating bacterial sexually transmitted infections (STIs), such as Chlamydia trachomatis. It facilitates the identification of new infected cases that can be treated through individual case management. PN also acts indirectly by limiting onward transmission in the general population. However, the impact of PN, both at the level of individuals and the population, remains unclear. Since it is difficult to study the effects of PN empirically, mathematical and computational models are useful tools for investigating its potential as a public health intervention. To this end, we developed an individual-based modeling framework called Rstisim. It allows the implementation of different models of STI transmission with various levels of complexity and the reconstruction of the complete dynamic sexual partnership network over any time period. A key feature of this framework is that we can trace an individual's partnership history in detail and investigate the outcome of different PN strategies for C. trachomatis. For individual case management, the results suggest that notifying three or more partners from the preceding 18 months yields substantial numbers of new cases. In contrast, the successful treatment of current partners is most important for preventing re-infection of index cases and reducing further transmission of C. trachomatis at the population level. The findings of this study demonstrate the difference between individual and population level outcomes of public health interventions for STIs.

Size matters: concurrency and the epidemic potential of HIV in small networks

Background

Generalized heterosexual epidemics are responsible for the largest share of the global burden of HIV. These occur in populations that do not have high rates of partner acquisition, and research suggests that a pattern of fewer, but concurrent, partnerships may be the mechanism that provides the connectivity necessary for sustained transmission. We examine how network size affects the impact of concurrency on network connectivity.

Methodology/Principal Findings

We use a stochastic network model to generate a sample of networks, varying the size of the network and the level of concurrency, and compare the largest components for each scenario to the asymptotic expected values. While the threshold for the growth of a giant component does not change, the transition is more gradual in the smaller networks. As a result, low levels of concurrency generate more connectivity in small networks.

Conclusions/Significance

Generalized HIV epidemics are by definition those that spread to a larger fraction of the population, but the mechanism may rely in part on the dynamics of transmission in a set of linked small networks. Examples include rural populations in sub-Saharan Africa and segregated minority populations in the US, where the effective size of the sexual network may well be in the hundreds, rather than thousands. Connectivity emerges at lower levels of concurrency in smaller networks, but these networks can still be disconnected with small changes in behavior. Concurrency remains a strategic target for HIV combination prevention programs in this context.

Weighting for sex acts to understand the spread of STI on networks

Human sexual networks exhibit a heterogeneous structure where few individuals have many partners and many individuals have few partners. Network theory predicts that the spread of sexually transmitted infections (STI) on such networks should exhibit striking properties (e.g. rapid spread). However, these properties cannot be found in epidemiological data. Current network models typically assume a constant STI transmission risk per partnership, which is unrealistic because it implies that sexual activity is proportional to the number of partners and that individuals have the same activity with each partner. We develop a framework that allows us to weight any sexual network based on biological assumptions. Our results indicate that STI spreading on the resulting weighted networks do not have heterogeneous-related properties, which is consistent with data and earlier studies.

Determinants of sexual network structure and their impact on cumulative network measures

There are four major quantities that are measured in sexual behavior surveys that are thought to be especially relevant for the performance of sexual network models in terms of disease transmission. These are (i) the cumulative distribution of lifetime number of partners, (ii) the distribution of partnership durations, (iii) the distribution of gap lengths between partnerships, and (iv) the number of recent partners. Fitting a network model to these quantities as measured in sexual behavior surveys is expected to result in a good description of Chlamydia trachomatis transmission in terms of the heterogeneity of the distribution of infection in the population. Here we present a simulation model of a sexual contact network, in which we explored the role of behavioral heterogeneity of simulated individuals on the ability of the model to reproduce population-level sexual survey data from the Netherlands and UK. We find that a high level of heterogeneity in the ability of individuals to acquire and maintain (additional) partners strongly facilitates the ability of the model to accurately simulate the powerlaw-like distribution of the lifetime number of partners, and the age at which these partnerships were accumulated, as surveyed in actual sexual contact networks. Other sexual network features, such as the gap length between partnerships and the partnership duration, could-at the current level of detail of sexual survey data against which they were compared-be accurately modeled by a constant value (for transitional concurrency) and by exponential distributions (for partnership duration). Furthermore, we observe that epidemiological measures on disease prevalence in survey data can be used as a powerful tool for building accurate sexual contact networks, as these measures provide information on the level of mixing between individuals of different levels of sexual activity in the population, a parameter that is hard to acquire through surveying individuals.

Circular stochastic fluctuations in SIS epidemics with heterogeneous contacts among sub-populations

The conceptual difference between equilibrium and non-equilibrium steady state (NESS) is well established in physics and chemistry. This distinction, however, is not widely appreciated in dynamical descriptions of biological populations in terms of differential equations in which fixed point, steady state, and equilibrium are all synonymous. We study NESS in a stochastic SIS (susceptible-infectious-susceptible) system with heterogeneous individuals in their contact behavior represented in terms of subgroups. In the infinite population limit, the stochastic dynamics yields a system of deterministic evolution equations for population densities; and for very large but finite systems a diffusion process is obtained. We report the emergence of a circular dynamics in the diffusion process, with an intrinsic frequency, near the endemic steady state. The endemic steady state is represented by a stable node in the deterministic dynamics. As a NESS phenomenon, the circular motion is caused by the intrinsic heterogeneity within the subgroups, leading to a broken symmetry and time irreversibility.

HIV transmissions by stage in dynamic sexual partnerships

Most models assessing relative transmissions during different progressive stages of human immunodeficiency virus (HIV) infection assume that infections are transmitted through instantaneous sexual contacts. In reality, however, HIV will often be transmitted through repeated sex acts during partnerships that form and dissolve at varying rates. We sought to understand how dynamic sexual partnerships would influence transmissions during different progression stages of HIV infection: primary HIV infection (PHI) and chronic stage. Using a system of ordinary differential equations with a pair approximation technique, we constructed a model of HIV transmission in a homogeneous population in which sexual partnerships form and dissolve. We derived analytical expressions for useful epidemiological quantities such as basic reproduction number and also did simulation runs of the model. Partnership dynamics strongly influence transmissions during progressive stages of HIV infection. The fraction of transmissions during PHI has a U-shaped relationship with respect to the rate of partnership change, where the minimum and maximum occur given partnerships of about 100 days and fixed partnerships, respectively. Models that assume instantaneous contacts may overestimate transmissions during PHI for real, dynamic sexual partnerships with varying (non-zero) durations.

Time for change? An economic evaluation of integrated cervical screening and HPV immunization programs in Canada

Many jurisdictions have implemented universal human papillomavirus (HPV) immunization programs in preadolescent females. However, the cost-effectiveness of modified cervical screening guidelines and/or catch-up immunization in older females in Canada has not been evaluated. We conducted a cost-utility analysis of screening and immunization with the bivalent vaccine for the Canadian setting from the Ministry of Health perspective. We used a dynamic model to capture herd immunity and included cross-protection against strains not included in the vaccine. We found that adding catch-up immunization to the current program would be cost-effective, and that combining catch-up immunization with delaying the age at which screening is first initiated could result in cost savings and net health gains.

A decade of modelling research yields considerable evidence for the importance of concurrency: a response to Sawers and Stillwaggon

In their recent article, Sawers and Stillwaggon critique the "concurrency hypothesis" on a number of grounds. In this commentary, I focus on one thread of their argument, pertaining to the evidence derived from modelling work. Their analysis focused on the foundational papers of Morris and Kretzschmar; here, I explore the research that has been conducted since then, which Sawers and Stillwaggon leave out of their review. I explain the methodological limitations that kept progress on the topic slow at first, and the various forms of methodological development that were pursued to overcome these. I then highlight recent modelling work that addresses the various limitations Sawers and Stillwaggon outline in their article. Collectively, this line of research provides considerable support for the modelling aspects of the concurrency hypothesis, and renders their critique of the literature incomplete and obsolete. It also makes clear that their call for "an end (or at least a moratorium) to research on sexual behaviour in Africa" that pertains to concurrency is unjustified.

When do sexual partnerships need to be accounted for in transmission models of human papillomavirus?

Human papillomavirus (HPV) is often transmitted through sexual partnerships. However, many previous HPV transmission models ignore the existence of partnerships by implicitly assuming that each new sexual contact is made with a different person. Here, we develop a simplified pair model—based on the example of HPV—that explicitly includes sexual partnership formation and dissolution. We show that not including partnerships can potentially result in biased projections of HPV prevalence. However, if transmission rates are calibrated to match empirical pre-vaccine HPV prevalence, the projected prevalence under a vaccination program does not vary significantly, regardless of whether partnerships are included.

Eigenvalue spectra of spatial-dependent networks

Many real-life networks exhibit a spatial dependence; i.e., the probability to form an edge between two nodes in the network depends on the distance between them. We investigate the influence of spatial dependence on the spectral density of the network. When increasing spatial dependence in Erdös-Rényi, scale-free, and small-world networks, it is found that the spectrum changes. Due to the spatial dependence the degree of clustering and the number of triangles increase. This results in a higher asymmetry (skewness). Our results show that the spectrum can be used to detect and quantify clustering and spatial dependence in a network.

[Determinants of condom use and heterosexual multiple sexual partnership in French Antilles and French Guiana]

BACKGROUND

After Subsaharan Africa, the Caribbean is the world's region most affected by HIV/AIDS. The French-American departments (Guadeloupe, Martinique, Guiana), FAD, are located in the heart of this region. Although lower than in other states of the Caribbean, AIDS incidence is much more higher than in France (up to 15 times more in Guiana). Transmission is mostly heterosexual. The frequency, particularly among men, of multiple sexual partnerships frequently taking place concurrently, and the persistence of this activity in older age, contribute to the level of the HIV epidemic and its characteristics. The purpose of this article is to identify, in the FAD, the determinants of condom use among persons with multiple sexual partners (either at last intercourse or during a concurrent relationship), taking into account the variety of multiple sexual partnership situations.

METHOD

Data are taken from an HIV/AIDS KABP survey, based on a probability sample of men and women aged 18 to 69 years, resident in FAD. In total, 3104 interviews were conducted by telephone in 2004: around 1000 in each department.

RESULTS

Among men and women who report two or more partners in the past five years, there is substantial heterogeneity in level of condom use at last intercourse, depending on the duration and type of the relationship: 73% of respondents reported condom use with a casual partner and 14% with a cohabiting partner. Men and women who were engaged in concurrent partnerships in the past five years were at higher risk of infection: 7% reported an STI versus 4% among those who had two or more partners, but not at the same time. Women, older persons, people with a lower level of education and those engaged in concurrent partnerships reported a lower level of condom use, thus increasing their vulnerability to HIV/AIDS.

CONCLUSION

Although HIV/AIDS prevention has increased among male and women engaged in multiple sexual partnerships, there is still a lack of consistent condom use in this population. These results highlight the need for more diversified prevention programs, taking into account sociodemographic factors and the diversity of situations involving multiple sexual partnership.

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.

Modeling HIV transmission risk among Mozambicans prior to their initiating highly active antiretroviral therapy

Understanding sexual behavior and assessing transmission risk among people living with HIV-1 is crucial for effective HIV-1 prevention. We describe sexual behavior among HIV-positive persons initiating highly active antiretroviral therapy (HAART) in Beira, Mozambique. We present a Bernoulli process model (tool available online) to estimate the number of sexual partners who would acquire HIV-1 as a consequence of sexual contact with study participants within the prior three months. Baseline data were collected on 350 HAART-naive individuals 18-70 years of age from October 2004 to February 2005. In the three months prior to initiating HAART, 45% (n = 157) of participants had sexual relationships with 191 partners. Unprotected sex occurred in 70% of partnerships, with evidence suggesting unprotected sex was less likely with partners believed to be HIV-negative. Only 26% of the participants disclosed their serostatus to partners with a negative or unknown serostatus. Women were less likely to report concurrent relationships than were men (21 versus 66%; OR 0.13; 95%CI: 0.06, 0.26). Given baseline behaviors, the model estimated 23.2 infections/1,000 HIV-positive persons per year. The model demonstrated HAART along with syphilis and herpes simplex virus type 2 (HSV-2) treatment combined could reduce HIV-1 transmission by 87%; increasing condom use could reduce HIV-1 transmission by 67%.

Invasive advance of an advantageous mutation: Nucleation theory

For sedentary organisms with localized reproduction, spatially clustered growth drives the invasive advance of a favorable mutation. We model competition between two alleles where recurrent mutation introduces a genotype with a rate of local propagation exceeding the resident's rate. We capture ecologically important properties of the rare invader's stochastic dynamics by assuming discrete individuals and local neighborhood interactions. To understand how individual-level processes may govern population patterns, we invoke the physical theory for nucleation of spatial systems. Nucleation theory discriminates between single-cluster and multi-cluster dynamics. A sufficiently low mutation rate, or a sufficiently small environment, generates single-cluster dynamics, an inherently stochastic process; a favorable mutation advances only if the invader cluster reaches a critical radius. For this mode of invasion, we identify the probability distribution of waiting times until the favored allele advances to competitive dominance, and we ask how the critical cluster size varies as propagation or mortality rates vary. Increasing the mutation rate or system size generates multi-cluster invasion, where spatial averaging produces nearly deterministic global dynamics. For this process, an analytical approximation from nucleation theory, called Avrami's Law, describes the time-dependent behavior of the genotype densities with remarkable accuracy.

Space and contact networks: capturing the locality of disease transmission

While an arbitrary level of complexity may be included in simulations of spatial epidemics, computational intensity and analytical intractability mean that such models often lack transparency into the determinants of epidemiological dynamics. Although numerous approaches attempt to resolve this complexity-tractability trade-off, moment closure methods arguably offer the most promising and robust frameworks for capturing the role of the locality of contact processes on global disease dynamics. While a close analogy may be made between full stochastic spatial transmission models and dynamic network models, we consider here the special case where the dynamics of the network topology change on time-scales much longer than the epidemiological processes imposed on them; in such cases, the use of static network models are justified. We show that in such cases, static network models may provide excellent approximations to the underlying spatial contact process through an appropriate choice of the effective neighbourhood size. We also demonstrate the robustness of this mapping by examining the equivalence of deterministic approximations to the full spatial and network models derived under third-order moment closure assumptions. For systems where deviation from homogeneous mixing is limited, we show that pair equations developed for network models are at least as good an approximation to the underlying stochastic spatial model as more complex spatial moment equations, with both classes of approximation becoming less accurate only for highly localized kernels.

Sexually transmitted disease epidemics in a natural insect population

1. The epidemiology of sexually transmitted diseases (STDs) in human and domesticated populations is well documented. However, there has been less study of STDs in natural populations. 2. We investigated STD dynamics in the model system involving a host from the most speciose group of animals: the insects. We investigated temporal variation in the prevalence of the sexually transmitted mite Coccipolipus hippodamiae on its ladybird host, Adalia bipunctata. 3. Field surveys over two seasons showed a repeated pattern of a profound epidemic in the overwintered cohort and a later prevalence decline. 4. In order to understand the key factors in the dynamics of this system we studied the phenology of the host and simulated parasite dynamics in the overwintered cohort using a model with within-sex homogeneity in mating rate and field-measured parameter values. The similarity of natural and simulation prevalence levels allowed us to carry out sensitivity analysis and hence to identify the key determinants of the dynamics. 5. The observed pattern of periodic extreme prevalence combined with system persistence probably results from time lags in host recruitment and widespread promiscuity. 6. Our findings improve our understanding of STDs in natural populations and illustrate the importance of examining seasonality and time delays in population dynamics in order to fully understand the characteristics of natural populations and their parasites.

HIV knowledge and risk behaviors among women in law enforcement in Bogota, Colombia: potential role as community educators

As HIV infection is increasing among women, evaluation, prevention, and education campaigns need to target this vulnerable population. Because of their frequent and accepted contact with members of the community, female law officers, if knowledgeable, could be well suited to provide information/education related to HIV/STD transmission. A survey of HIV/AIDS knowledge and risk behaviors was administered to 120 law enforcement women (LEW) and 60 women from the general population (GPW) in Bogotá, Colombia. LEW indicated a very high (90%) understanding of basic HIV knowledge. Although most (52%) of the LEW did not report high-risk behaviors, 29% indicated having unprotected sex during menses, and 17% had unprotected anal sex. This contrasts, however, with GPW, who were of similar age, but had a significantly higher prevalence (73%) of risky behaviors (P=.004). Moreover, 52% of the GPW reported having unprotected anal sex, and approximately half of this group (55%) indicated having unprotected sex during menses. Alcohol and drug users were also more prevalent in the GPW: 14% frequently used alcohol and 3% inhaled drugs during sexual encounters, contrasted to 2% of LEW reporting alcohol use. GPW were four times more likely than LEW, to engage in high-risk sexual practices [95% confidence interval (CI)=1.9-10.4, P=0.034]. Multivariate analyses indicated that alcohol and/or drug use were significantly associated with high-risk sexual practices [odds ratio (OR)=4.7, 95% confidence intervals (CI)=1.3-18.4, P=.02). Improved educational HIV/AIDS programs are needed, particularly for women in the general population, who use alcohol/drugs during sexual encounters, which account, at least in part, for their high-risk behaviors. Women in law enforcement, who appear knowledgeable and exhibit safer behaviors, could be useful educators for GPW. Because of their professional role in the community, training for LEW in HIV/AIDS education/prevention programs should be considered.

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.

Cluster approximations for epidemic processes: a systematic description of correlations beyond the pair level

The spread of a virus is an example of a dynamic process occurring on a discrete spatial arrangement. While the mean-field approximation reasonably reproduces the spreading behaviour for topologies where the number of connections per node is either high or strongly fluctuating and for those that show small-world features, it is inaccurate for lattice structured populations. Various improvements upon the ordinary pair approximation based on a number of assumptions concerning the higher-order correlations have been proposed. To find approaches that allow for a derivation of their dynamics remains a great challenge. By representing the population with its connectivity patterns as a homogeneous network, we propose a systematic methodology for the description of the epidemic dynamics that takes into account spatial correlations up to a desired range. The equations that the dynamical correlations are subject to are derived in a straightforward way, and they are solved very efficiently due to their binary character. The method embeds very naturally spatial patterns such as the presence of loops characterizing the square lattice or the tree-like structure ubiquitous in random networks, providing an improved description of the steady state as well as the invasion dynamics.

Frequency-dependent incidence in models of sexually transmitted diseases: portrayal of pair-based transmission and effects of illness on contact behaviour

We explore the transmission process for sexually transmitted diseases (STDs). We derive the classical frequency-dependent incidence mechanistically from a pair-formation model, using an approximation that applies to populations with rapid pairing dynamics (such as core groups or non-pair-bonding animals). This mechanistic derivation provides a framework to assess how accurately frequency-dependent incidence portrays the pair-based transmission known to underlie STD dynamics. This accuracy depends strongly on the disease being studied: frequency-dependent formulations are more suitable for chronic less-transmissible infections than for transient highly transmissible infections. Our results thus support earlier proposals to divide STDs into these two functional classes, and we suggest guidelines to help assess under what conditions each class can be appropriately modelled using frequency-dependent incidence. We then extend the derivation to include situations where infected individuals exhibit altered pairing behaviour. For four cases of increasing behavioural complexity, analytic expressions are presented for the generalized frequency-dependent incidence rate, basic reproductive number (R0) and steady-state prevalence (i infinity) of an epidemic. The expression for R0 is identical for all cases, giving refined insights into determinants of invasibility of STDs. Potentially significant effects of infection-induced changes in contact behaviour are illustrated by simulating epidemics of bacterial and viral STDs. We discuss the application of our results to STDs (in humans and animals) and other infectious diseases.

Monogamous networks and the spread of sexually transmitted diseases

Patterns of sexual mixing and heterogeneity in the number of sexual partners can have a huge effect on the spread of a sexually transmitted disease (STD). The sexual mixing network identifies all partnerships within a population over a given period and is a powerful tool in the study of such infections. Previous models assumed all links within the network to be concurrent active partnerships. We present a novel modelling approach in which we adapt the notion of a sexual contact network to a monogamous population by allowing the nature of the links to change. We use the underlying network to represent potential sexual partnerships, only some of which are active at any one time. Thus serial monogamy can be modelled while maintaining the patterns of mixing displayed by the population.

Contact tracing and disease control

Contact tracing, followed by treatment or isolation, is a key control measure in the battle against infectious diseases. It is an extreme form of locally targeted control, and as such has the potential to be highly efficient when dealing with low numbers of cases. For this reason it is frequently used to combat sexually transmitted diseases and new invading pathogens. Accurate modelling of contact tracing requires explicit information about the disease-transmission pathways from each individual, and hence the network of contacts. Here, pairwise-approximation methods and full stochastic simulations are used to investigate the utility of contact tracing. A simple relationship is found between the efficiency of contact tracing necessary for eradication and the basic reproductive ratio of the disease. This holds for a wide variety of realistic situations including heterogeneous networks containing core-groups or super-spreaders, and asymptomatic individuals. Clustering (transitivity) within the transmission network is found to destroy the relationship, requiring lower efficiency than predicted.

Using network models to approximate spatial point-process models

Spatial effects are fundamental to ecological and epidemiological systems, yet the incorporation of space into models is potentially complex. Fixed-edge network models (i.e. networks where each edge has the same fixed strength of interaction) are widely used to study spatial processes but they make simplistic assumptions about spatial scale and structure. Furthermore, it can be difficult to parameterize such models with empirical data. By comparison, spatial point-process models are often more realistic than fixed-edge network models, but are also more difficult to analyze. Here we develop a moment closure technique that allows us to define a fixed-edge network model which predicts the prevalence and rate of epidemic spread of a continuous spatial point-process epidemic model. This approach provides a systematic method for accurate parameterization of network models using data from continuously distributed populations (such as data on dispersal kernels). Insofar as point-process models are accurate representations of real spatial biological systems, our example also supports the view that network models are realistic representations of space.

Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases

A wide range of communicable human diseases can be considered as spreading through a network of possible transmission routes. The implied network structure is vital in determining disease dynamics, especially when the average number of connections per individual is small as is the case for many sexually transmitted diseases (STDs). Here we develop an intuitive mathematical framework to deal with the heterogeneities implicit within contact networks and those that arise because of the infection process. These models are compared with full stochastic simulations and show excellent agreement across a wide range of parameters. We show how such models can be used to estimate parameters of epidemiological importance, and how they can be extended to examine the effectiveness of various control strategies, in particular screening programs and contact tracing.

Mathematical biology and medical statistics: contributions to the understanding of AIDS epidemiology

Some of the many ways in which mathematical biology and statistics have been used in investigating the acquired immunodeficiency syndrome (AIDS) epidemic are reviewed. Aspects of the spread of the disease via social and sexual networks are discussed. The different kinds of data involved are critically compared. Some studies of the incubation period are briefly reviewed and some comments made on the role of adherence to therapy.