Injecting drug users (IDUs) are at the vanguard of the HIV epidemic in large parts of the world, and in many countries explosive epidemics among IDUs may be driving sexual transmission to the wider population [1–5]. Especially in Asia, injecting drug use has paved the way for HIV into other risk groups and the general population, and the geographic spread of HIV has been associated with major drug-trafficking routes [6,7]. Also, injecting drug use has been driving the rapid spread of HIV in Eastern Europe following the break-up of the Soviet Union [8,9]. The epidemiology of HIV infection and other drug-related infectious diseases in IDUs is, however, difficult to study and understand. IDUs are hidden populations for whom entry to and exit from the population are hard to define and measure. The transmission of infectious diseases in IDUs is linked to disease transmission in other high-risk groups (men who have sex with men, sex workers). In addition, the epidemiology of infectious diseases interacts with the epidemiology of drug use and injecting. These complex mechanisms lead to diverging developments of HIV prevalence in IDU populations in different areas of the world, the reasons for which are not well understood. For example, in the European Union (EU), there are countries with continuous near-zero HIV prevalence among IDUs, whereas others recently experienced large outbreaks similar to those observed in other parts of the world [10–14]. At the same time, the prevalence of chronic hepatitis C virus (HCV) infection is found to be high across different countries and populations and IDU is now a main transmission route [15,16], which is likely to make a substantial impact on healthcare. Mathematical modeling can help to construct a consistent framework in order to identify and analyze the driving forces for further spread of infection and the potential impact of intervention.
Recently, new developments have added valuable information for conducting modeling studies. The increasing implementation of second generation HIV surveillance  is starting to deliver information on risk behavior among IDUs and their distributions and time trends [3,18,19]. Novel data on the distribution of different HIV and HCV subtypes and genotypes obtained from molecular typing in combination with epidemiological information is leading to a better understanding of contact and mixing patterns among different subpopulations of IDU [20,21]. Interventions are also changing; HIV and hepatitis C treatment have greatly improved [22–25] and so have the coverage and design of harm-reduction measures [e.g. substitution treatment, needle and syringe programs (NSP)] in large parts of the world [26,27] although IDUs often still have less access to viral treatment [25,28]. HIV counseling and testing is likely to be strongly scaled up following recent recommendations [29–31], and new tests are being developed to detect recently acquired HIV infections, which may enable a more direct measurement of HIV incidence and offer opportunities for effective contact tracing .
In this review, after giving a brief overview of what has been done in mathematical modeling of HIV and HCV in IDUs so far, we outline the most important new developments concerning HIV and HCV in IDUs and discuss how mathematical and statistical modeling can help to investigate their importance in epidemiology and intervention.
Mathematical modeling studies for drug-related infectious diseases
Already early on in the HIV epidemic among IDUs, mathematical modeling has had an impact on public health policy making by changing the way of thinking about needle exchange programs. In the late 1980s, an NSP was implemented in the city of New Haven, Connecticut, which was then heavily hit by the HIV epidemic among IDUs. For evaluating the impact of the program, Kaplan et al.[33,34] designed a system of tracking and collecting used syringes and testing them for HIV. Using a mathematical model, they could provide an estimate for the decrease in HIV incidence in the IDU population of New Haven, attributable to the needle exchange program. Their model was based on the idea of viewing needles as the vectors that carry the pathogen from infected to susceptible person, comparable with the earliest models for malaria transmission by Ross . With results from the models used, they were able to convince policy makers to continue the needle exchange program and to extend it to other cities in the US. The modeling work by Kaplan et al.[33,34] is exemplary in the way it combines simple modeling tools with ingenious ideas, empirically based parameter estimates and policy-relevant questions. Other work has continued in this direction; for example, on the basis of the model by Kaplan et al. Greenhalgh et al.[36–38] developed a mathematical framework for analyzing the spread of HIV in a population of IDUs, also taking into account variable infectivity of infected IDUs.
The need for good empirically based parameter estimates was highlighted in the studies by Blower et al.[39,40], who introduced a model to study the contribution of sharing injection equipment and sexual contact in the spread of HIV in New York City. The authors showed that the first epidemic wave was driven by transmission through needle sharing among IDUs. After a period of leveling off, a second wave followed wherein HIV had spread through sexual transmission into the larger community of noninjecting drug users and sex partners of IDUs. The observed stabilization in the number of new AIDS cases in New York City was, therefore, not necessarily a consequence of changes in risk behavior but could just as well be explained by the intrinsic transmission dynamics of the epidemic.
Earlier, the importance of describing the social and injecting contact networks of IDUs was recognized. Although a distinction was already made between sharing needles with strangers or with friends , only individual-based modeling approaches [41,42] were able to model contact networks of IDUs in greater detail. The latter study showed that preventing needle sharing with strangers or in anonymous settings and targeting the prevention in beginning injectors could be effective in reducing HIV incidence. In contrast, increased HIV testing and behavior change in those diagnosed HIV positive did not prove to be effective in the model due to a high-transmission probability during primary HIV infection. Results from empirical studies on IDUs' social and injecting networks suggested that network structure might be an important factor to explain the observed stability of HIV prevalence levels among IDUs in New York City . Similarly, for HCV infection among IDUs in Glasgow, a stochastic simulation model showed that confining needle sharing to only one partner can greatly reduce the spread of infection as compared with sharing among randomly chosen partners .
Mathematical models have also been used to investigate the role of IDUs in driving HIV transmission in the transition from concentrated to generalized epidemics [9,45]. Although large uncertainties in parameter values prohibit quantitative predictions of the influence of IDUs on the expansion of HIV epidemics, the role of IDUs as sentinel populations and as core groups in rapid and extensive spread of HIV can be investigated. The extent of contact of IDUs with bridge populations such as commercial sex workers determines the rate of spread into the general population, but a self-sustaining generalized epidemic is only possible if sufficiently high-risk behavior in the general population leads to continuous heterosexual transmission . Models have also been used to estimate the effectiveness of interventions aimed at preventing the spread of HIV from IDUs to other risk groups, especially in Asian countries [47,48]. These studies have shown the key role of IDUs in the emerging HIV epidemic in Asia and underline the need for effective prevention targeted at IDUs in these countries.
On the basis of a proposed disease progression model , Vickerman et al.[50,51] recently developed a model for the analysis of the impact of NSP on the epidemiology of HIV and HCV infection among IDUs. Using data from London, they showed that a moderate coverage of syringe exchange could have a substantial impact on HIV, whereas for a reduction in HCV prevalence, large decrease in sharing frequency and thus very high coverage is necessary. Law et al.[52,53] used a mathematical model to project the sequelae of hepatitis C infection in a cohort of IDUs in Australia over a period of 20 years. The authors also estimated the number of quality-adjusted life years lost to chronic HCV infection in 2001 mainly due to infection in IDUs, showing that effective prevention of HCV infection could have a large impact on population health. A comparable study  of the future burden of disease caused by hepatitis C among IDUs in Scotland also supports the conclusion that prevention of infection and better treatment of former and current IDUs can have a large impact on future morbidity and mortality of hepatitis C.
Capture–recapture and other statistical methods have been used to estimate the size of (injecting) drug user populations [55–57]. Statistical models were also used to estimate the incubation time distribution for HIV until AIDS symptoms appeared , which served as important information for use in back-calculation methods to estimate the incidence of HIV infection [59–61]. Back-calculation and other lag-correction techniques have also been applied to estimate the prevalence and incidence of IDU using drug treatment and mortality data [62–65]. Statistical models have also been developed to estimate the force of infection for HCV among IDUs depending on the time since first injection . In a similar approach, the contribution of different risk behaviors to the force of infection of HCV was evaluated using data from three IDU populations in Belgium . The study highlighted the risks of HCV transmission by injecting paraphernalia other than syringes, which limits the possible impact of NSPs on HCV transmission.
Mathematical modeling is increasingly employed in the context of cost-effectiveness analysis of infectious disease control to assess indirect effects of intervention attributable to changes in the prevalence and force of infection. Such an approach was taken by Pollack  to analyze the cost-effectiveness of NSP in preventing HCV infection in IDUs, and similarly by Vickerman et al. and Kumaranayake et al. for estimating the cost-effectiveness of NSP in preventing HIV in Ukraine and Belarus, respectively. For HIV, harm-reduction interventions were found to be cost effective, even in the cases where coverage was not sufficient to substantially reduce HIV prevalence; however, in the case of HCV, the intervention was predicted to have little impact on prevalence and incidence, suggesting the need for more comprehensive harm-reduction models.
Emerging research issues for mathematical modeling
The knowledge on the epidemiology of infectious diseases in IDUs and options for prevention has increased in recent years, and policy makers have new and different decisions to take than just a decade ago. The plethora of new research questions that have arisen with these developments opens up new opportunities for application of mathematical models (Table 1).
Several methods have been developed to distinguish recent HIV infection from established chronic infection mainly for estimating HIV incidence from cross-sectional data . These methods rely on various mechanisms of interaction of the virus with the host immune system [72–74]. Efficient pooling algorithms have made it possible to apply nucleic acid amplification testing for acute infection in routine surveillance settings . The possibilities of identifying acute and early HIV infection have raised the question of whether highly active antiretroviral therapy (HAART) should be started earlier in an attempt to influence further prospects for an infected person by lowering the viral set point or prolonging the period of high CD4 cell counts. No decisive clinical trials have yet been conducted to provide a conclusion for the best treatment strategy , but it is expected that testing for recent infection and new treatment options will have an impact on the epidemiology of HIV in the future . Mathematical modeling can help to weigh opposing effects of treatment such as increased survival and lowering of the viral load against each other and predict the likely effects of changes in the treatment coverage in the population.
Testing for acute or recent HIV infection opens up the possibility for investigating the role of primary HIV infection in forward transmission . The evidence for the importance of primary infection comes mainly from studies among men who have sex with men although its role for transmission in IDU populations does not seem to have been investigated. Similarly, one can ask whether acute infection with HCV plays an important role in the forward transmission of the infection among IDUs, especially among beginners .
For studying the transmission of viral infections among and between IDU risk groups, molecular typing in combination with epidemiological investigations of individual cases or small clusters has proved to be a useful tool. Molecular epidemiological studies on the spread of HCV among IDUs in Amsterdam have provided evidence of viral exchange with countries where other genotypes are circulating , thereby indicating changes in the composition of drug-user populations over time. In a Belgian study , different genotypes could be attributed to injecting compared with noninjecting drug users, and tattooing emerged as a risk factor for the transmission of another genotype. The distribution of genotypes among and between subpopulations is closely related to the contact networks in the IDU population and provides insight on how subpopulations are connected with each other. Ground-breaking research on contact networks in IDU populations had been conducted in New York City in the 1990s  and in the Colorado Springs Study [77,78], and network structure has been linked to risk behavior characteristics [79,80] and intervention success [81,82]. Mathematical modeling is increasingly used to study the relationship between the contact network structure and the spread of genotypes of HIV and HCV, providing potentially important insights for targeting intervention and for prevention of the transmission of treatment-resistant genotypes [83,84].
Large-scale surveillance allows evaluation of the impact on the population level of public health policies. In recent years, the concept of ‘second generation HIV surveillance (SGS)’ has been introduced, which combines HIV/AIDS case-reporting surveillance with repeated (sero) behavioral surveys in IDUs and includes other sources of information such as data on sexually transmitted infections (STIs) . It has been proposed to include additional indicators of risk and access to prevention services, which are specific to high-risk groups. Examples for additional data to be collected might be the coverage of methadone maintenance and NSP or HCV prevalence and notifications as population level indicators for injecting risk behavior . Increasing application of SGS [3,18,19] will improve the availability of risk group-specific and country-specific behavioral data and thus enhance the possibilities to study intervention options using statistical and mathematical modeling.
The epidemiology of HCV is complicated by indications that immunity might not be complete such that reinfection and superinfection can occur [87–90]. Waning immunity and reinfection may lead to dynamic phenomena that make interpretation of epidemiological observations difficult; for example, it may lead to a steep increase in the endemic prevalence for small changes in the transmission probability . If substantiated, this could complicate the interpretation of the relationship between HCV prevalence and injecting risk behavior. Mathematical modeling can be used for investigating the impact of acute or active HCV infection on the epidemiology of HCV. However, in most epidemiological studies, information is available only about the prevalence of HCV antibodies and not about active HCV infection. In view of the similarities and differences in the epidemiology of HIV and HCV, studying the epidemiology of coinfection with these viruses can provide potential and additional insight into the transmission dynamics of both infections separately. Not only are monoinfected IDUs at a higher risk of contracting a second infection than noninfected IDUs, but the infections influence each other through the immune system. HIV infection leads to higher HCV viral loads and faster progression to end-stage liver disease, whereas HCV infection may render HIV treatment more difficult. However, the viral interactions may be HCV genotype and gender specific, and the findings on the effects of HCV infection on HIV progression have not been consistent [92,93].
The possibility of generalization of insights from behavioral surveillance or epidemiological studies among IDUs depends strongly upon how participants are recruited. Being a hidden population, recruitment of IDUs has always been fraught with uncertainties concerning the representativeness of a sample. Recently, methods based on snowball sampling [94–97] have been developed further into the so-called respondent-driven sampling (RDS), in which participants in the sample are paid to recruit some of their contacts as new participants following fixed rules about the number of contacts and how they are to be recruited [98,99]. In theory, by using this method of recruitment, the composition of subsequent waves of participants converges to a distribution that reflects the contact patterns in the population. Mathematically, this procedure is similar to the concept of next generation matrix, which describes transmission within and between subgroups of the population . Therefore, samples from RDS could be used to investigate properties of contact and infection networks. RDS has been compared with more conventional sampling methods in IDUs with no major differences being found concerning HIV prevalence and risk behavior . Problems with this method, however, have already been encountered, for example, in settings where social networks are very weakly connected and stigmatization and control levels are high . Further validation of RDS still seems necessary as estimates of the prevalence of infectious diseases derived from RDS can be sensitive to model assumptions .
The availability of specific HIV prevention measures for IDUs such as methadone maintenance therapy and NSPs has highly improved in some regions and countries of the world, for example, in the EU [27,104] and, very recently, in China [105,106]. Evidence regarding the effectiveness of these services is rapidly accumulating [107–110] although some discussions still continue . Properly dosed, long-term methadone treatment has been found to be a central protective factor in preventing HIV infection through the reduction of opioid use and injecting . Other interventions, including not only the needle and syringe exchange or distribution programs (NSPs and supervised injection facilities [108,109], but also HIV counseling and testing, information, education and promotion of condom use, HAART and antiviral treatment and vaccination against hepatitis infection, community outreach and referral and, especially, a strong support for these interventions by national policies, and integrated, multidisciplinary, service delivery are likely important [26,110]. Expanding provider-initiated HIV testing and counseling in health services for most-at-risk populations are being considered in recent guidelines [29–31]. In the EU, it has been recommended that a voluntary medical examination and testing for a number of infectious diseases be a part of a routine annual offer for IDUs . Following changes in drug use patterns, including shifts from heroin to prescribed opioids such as buprenorphine, in some countries the recruitment of new IDUs has declined substantially, leading to a gradually declining prevalence of (and an ageing cohort of) IDUs in the population although in other countries this has not been the case .
For viral hepatitis the effectiveness of substitution (e.g. methadone) treatment is much less clear than in the case of HIV, although modeling suggests that a high coverage of the IDU population with methadone maintenance treatment, in combination with other harm reduction measures, might turn methadone maintenance treatment cost-effective . This seems to be confirmed by recent findings from the Amsterdam cohort studies  showing that methadone maintenance treatment or needle exchange alone did not reduce HCV (nor HIV) incidence among IDUs, but those IDUs who were using both measures were strongly protected against both infections. In general, the higher transmissibility of HCV as compared with HIV enables the virus to establish itself on a high-endemic prevalence at much lower risk behavior levels and therefore requires higher intervention efforts for reaching substantial prevalence reductions .
Although several studies have looked at the effectiveness of individual prevention measures, few have yet looked at their potential combined effects . Modeling is a tool for estimating the cost-effectiveness of different intervention mixes such as the combination of opioid maintenance treatment, needle exchange and HIV/HCV testing. Modeling may also be used to estimate the coverage needed to reach an optimum intervention impact . It might help clarify the potential effects of migration and/or of changing demographic structure , or help investigating the contribution of HIV transmission among IDUs in prisons . To ensure that modeling delivers useful results about the impact of drug use interventions, it is important that good quantitative information becomes available about the fractions and frequencies of IDUs' sharing needles and how these risk behaviors change with increasing availability of clean syringes. However, there is thought to be a considerable bias around basic behavioral data such as the proportion of IDUs' that share syringes and the frequency with which they share. On the contrary, self-reported behavioral data have been found to be sufficiently reliable and valid to provide descriptions of drug use and related problems and behaviors  and are highly important for parametrizing mathematical models .
Challenges for mathematical modeling
The developments sketched above concerning epidemiology, diagnosis, treatment and other interventions pose numerous questions where mathematical modeling can be used as a tool for generating and testing hypotheses about the relationship between effects on the individual and the population levels. To answer these questions adequately, mathematical modelers face the challenge of developing new methodology for analysis and of combining knowledge from different fields of research into a common framework.
Mathematical modeling of HIV and HCV among IDUs has been based on a range of different approaches from simple deterministic modeling with mass action terms to describe disease transmission up to complex individual-based models that take details of local contact structure into account. All models face the challenge of incorporating the knowledge about transmission and clinical course of infection that is available and at the same time dealing with large uncertainties due to gaps in knowledge. Similarly, detailed information is available about injecting behavior and networks, but there is no straightforward way of incorporating this information into a model. Defining an adequate model structure and parametrizing the model based on the available data is therefore a daunting job. Better statistical methods to validate complex model structures are urgently needed and comparing different models with respect to their performance and outcome is necessary. In Vickerman et al.  it was shown that alternative model parametrizations can well describe the same observed data, and that lack of knowledge makes it impossible to decide which parameterization best describes reality. Imperative for a successful application of mathematical modeling for gaining epidemiological insight is the continuous interaction between mathematical modelers and epidemiologists whereby, modelers generate hypotheses that can be tested by epidemiological studies, which in turn inform and improve mathematical models.
Mathematical modeling based on information from molecular typing of pathogens is still in its infancy. One line of research that was developed by Pybus et al. uses the phylogenetic tree of pathogens to infer the population size of the pathogen population over a time spanning the evolution of different strains. Also, the numerical value of the basic reproduction number can be estimated from the phylogenetic and demographic information therefore giving a quantitative estimate for the effort needed for reaching given intervention aims. The methods were applied to analyze the introduction and growth of hepatitis C in the population of IDUs and to estimate the transmission potential of different HCV subtypes . Similarly, only some isolated pieces of work up to now have approached the question of how contact structure influences the dynamics of pathogen competition and spread. Recently, Gordo and Campos  investigated the genetic variation of a pathogen spreading in a metapopulation of hosts connected by a small world network. Eames and Keeling  showed that competing pathogen strains can coexist on one host population if there is a strong heterogeneity in contact patterns allowing the evolutionary adjustment of a pathogen strain to a subgroup with specific contact behaviors, as has been empirically confirmed in . Those modeling results have yet to be tested on epidemiological data for evaluating their explanatory value for the observed distribution of genotypes across various risk groups. Also, these modeling approaches have to be developed further to account for details of mixing patterns and contact behavior of IDUs in order to understand how genotypes spread through IDU populations.
New developments in social network theory have led to statistical approaches to analyze and model contact network structure. The so-called scale free networks describe contact patterns where some individuals have a large number of contacts whereas the majority has only few contacts– a situation that resembles syringe-sharing patterns among IDUs. It was argued for sexually transmitted infections that the scale-free property of the sexual contact network precludes successful intervention against STI, because of the lack of an epidemic threshold . Similar arguments might hold for IDUs networks' where large heterogeneity in numbers of sharing partners has also been observed [50,70]. Other more statistical approaches in social network theory aim at capturing social network structure at the micro level by describing probability distributions of network structures such as triads and stars . Such a statistical approach makes it possible to simulate networks based on empirical data using Markov Chain Monte Carlo algorithms . These developments open new roads in analyzing the effects of network structure on HIV transmission in IDU populations presuming that empirical data about micro-structure of IDU contact networks can be collected.
The task for mathematical modeling is to promote understanding of the intricate dynamic relationship between contact networks and risk behavior, biological properties of viral genotypes and various stages of infectivity. A better understanding of these mechanisms is fundamental to optimize targeting of intervention, estimating the intervention effort needed for substantial prevalence reductions and minimizing the danger of transmission of treatment-resistant strains. Finally, sound mathematical modeling is a prerequisite for a state-of-the-art cost effectiveness analysis, a potential cornerstone but yet underused tool for policy decision-making.
New developments in the diagnosis of early HIV infection, in the treatment of HIV, HBV and HCV infections, in the design of intervention for drug use, in surveillance and in the possibilities of molecular typing of pathogens are changing the questions and decisions for public health policy makers who deal with drug-related infectious diseases. At the same time, more and detailed information is becoming available about epidemiological determinants and transmission routes of those pathogens in IDU populations. Modeling has much to offer for solving urgent policy-relevant questions, and a strong structural effort in terms of funding and providing research opportunities is necessary to make better use of these possibilities. Modeling studies that are to contribute to designing intervention and treatment policy have to be based on a close interdisciplinary collaboration of mathematical modelers, statisticians, epidemiologists and behavioral scientists. Such a collaborative network was set up in 2006 in Europe with support of the European Monitoring Centre for Drug and Drug Addiction (EMCDDA) and has led to the formation of an interdisciplinary study group for modeling drug-related infectious diseases.
The present work was conducted as part of the EMCDDA project Coordination of a working group to develop mathematical and statistical models and analyses of protective factors for HIV infection among injecting drug users (CT.06.EPI.205.1.0). We thank Ralf Reintjes for helpful comments and critical reading of the manuscript. We thank Heiko Jahn for his help with the literature research. We thank the members of the European study group for mathematical modeling and epidemiological analysis of drug-related infectious diseases for many stimulating discussions.
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