Anderson, Roy M. PhD, FRS; Garnett, Geoffrey P. PhD, BSc
A CENTRAL ROLE of mathematical models in the study of the epidemiology and control of sexually transmitted diseases (STDs) is to further knowledge of the interplay between the variables that determine the typical course of infection within an individual, and those that determine the pattern of infection in the community.1 In view of the successes achieved by interdisciplinary research that combines theory, experiment, and observation in the physical and engineering sciences, it is surprising that many people still question the potential usefulness of mathematical studies in biomedical fields. Sensibly used mathematics is no more and no less a way of thinking about issues and relationships between variables and processes in a precise way, using a language that helps traverse the barriers often imposed by different languages and their use by different individuals. Simple models appeal to the theoretician because of analytical tractability. However, they typically attract much criticism from the experimentalist or observational scientist because they ignore a great deal of known biological complexity. Starting simply and slowly adding complexity is a similar approach to that adopted by the experimental biologist, where one or few variables are allowed to change in an experimental design while others are held constant. The objective in both cases is to fully understand the contribution of one or few variables to an observed pattern, either acting alone or in combination. Mathematical models are wonderful tools to explore the influence of different factors on observed epidemiologic patterns in a precise and controlled manner. In this sense, they act as a template for learning, even though the degree of complexity explored may be far removed from reality.
No one doubts that the real world is full of much complexity, but the task of the scientist is to try and dissect those influences that play the major role as determinants of recorded pattern and its change over time or through space. In this endeavor, mathematics is a powerful tool that, when used sensibly, can provide much insight and great precision. In biomedical research, there is a slow dawning of realization that quantitative tools will be essential to unravel the detail lying within the ever‐increasing torrent of information generated by the new molecular tools of measurement and observation. The term bioinformatics is the topical word to describe a heterogeneous collection of needs‐it embraces mathematics, statistics, and software development plus a variety of other disciplines. In the postgenomic era, repetitive whole‐pathogen sequencing and the use of DNA chips to study variability at precise locations on the genome will be part of everyday epidemiologic study. Statistical, mathematical, and computational tools will be key to analysis and interpretation. The threshold of this era has arrived; but, in many respects we are ill prepared to tackle the tasks ahead because of widespread mathematical and statistical illiteracy in today's biomedical research community. In part, this is understandable given the lack of practical relevance of much of the mathematical biology literature. In addition, there has been a tendency to publish theoretical studies in obscure and little‐read biomathematical journals instead of in the mainstream biomedical literature. Furthermore, mathematical models often soar free from the constraints of reality and data, with the attention of the theoretician placed on the elegance of the formulation and analysis as opposed to the relevance of the model to real biological systems. However, these problems should not detract from the urgent need to encourage mathematical approaches to the study of biological systems, which almost without exception consist of many variables interacting in highly non‐linear ways. Sensibly applied, mathematics is a powerful tool for the dissection and analysis of these complex nonlinear systems.
Infectious disease epidemiology in general, and STD in particular, are fields of medicine in which mathematical models have become an integral part of current research. This is in part because of the highly nonlinear pattern of most infectious disease epidemics in naïve populations. They tend to be bell shaped in character and in many situations recurrent in form (e.g., childhood viral infections such as measles). Repeated waves of infection and associated disease may sweep through defined populations, often with intervals of many years between epidemic peaks superimposed over smaller‐scale seasonal cycles in incidence. In seeking explanations for such recurrent patterns, simple epidemic theory based on deterministic or stochastic mathematical models has been of great help. Indeed, for some infections even simple models have the capability of predicting future trends.
To date, models for the sexually transmitted infections have been used more as props to understanding and as tools to explore “what if” questions in the design of intervention programs rather than predictive methods. This brief report highlights the past and current uses of mathematical models in STD research and comments on future directions and needs.
The Use of Mathematical Models
Mathematical models in STD epidemiology have many different uses. A key need is to define the aim or aims of model development clearly before construction and analysis begins. Far too often in the current literature, the purpose of model development is poorly defined. Specification of aim helps to decide on the type of model needed, and what methods are most appropriate in analysis and investigation of model properties and predictions. Of particular importance is the definition of key variables plus parameters, and the concomitant requirements for comparison of prediction with observed pattern. Mathematical models can serve a number of purposes. The most important are as follows:
* to delineate the basic principals and processes underlying transmission;
* to determine what needs to be measured to both interpret epidemiological pattern and assess the impact of defined interventions;
* to help define what are the most important determinants of observed pattern;
* to help design and evaluate different interventions; and
* to test well‐defined hypotheses.
Sensibly used mathematics provides a precise language for model construction and for the delineation of the assumptions made. Once the basic framework is constructed, it is easy to determine what needs to be measured to assign parameter values and to compare prediction with observation. Measurement may necessitate experimentation, intervention, or observation, perhaps by surveillance. Through analytical and numerical methods, the behaviors of the system under different assumptions can be evaluated and predictions compared with observed trends. Formal sensitivity analyses, based on both the alteration of model structure and changing parameter assignments, can be used to assess how important key processes and parameters are as determinants of observed trends. With modern computational facilities many different structures and parameter assignments can be explored quickly and efficiently. This is of particular importance when key parameters or variables are difficult to measure, or where there is much uncertainty about the biological details governing the course of infection in the patient and the determinants of transmission between patients. A further advantage of abundant computational power on the desktop is the ability to develop complex simulation models that capture individual behaviors in a large population. There are, of course, many dangers associated with enormous computational power linked to the temptation to construct ever more‐complex models without any sharp analytical understanding of how specific processes or parameter assignments influence outcome. Ideally, the use of complex “individual behavior” simulation models should proceed hand‐in‐hand with analytical studies of simpler templates to aid interpretation.
A Brief History
The use of mathematical models in infectious disease epidemiology dates back to the work of Daniel Bernoulli, who in 1760 estimated the improvement in life expectancy resulting from variolation to protect from smallpox infection.2 Further development of a robust theoretical framework to describe the transmission dynamics of infectious diseases had to wait until the early 20th century, when Hamer3 recognized the importance of the rate of contact between susceptible and infected persons as a key determinant of the net rate of transmission. The development of a so‐called “mass‐action theory” by Hamer, in which the product of the density of susceptible and infected persons scaled by a transmission coefficient defines net transmission, underpins much of current theory in infectious disease epidemiology. Although the work of Hamer was based on a discrete time framework, Ronald Ross similarly explored the “mass‐action” assumption using a continuous time template based on differential equations.4 Despite these early developments in epidemic theory, it is only during the past 30 years that a mathematical framework has been developed for sexually transmitted infections. Following from some early applications of mathematical models to describe the transmission dynamics of gonorrhea,5–8 two seminal publications by Yorke and Hethcote reviewed and developed the subject.9,10 These publications, which were exemplary because of their grounding in data and their attempt to address issues of interest to health workers, introduced a number of concepts that remain important in STD epidemiology today. These include the notion of an “infectee” or “reproductive number” as a measure of transmission success, the importance of asymptomatic infections in maintaining endemic infection, and the role of a “core group” of individuals with high rates of sexual partner change in maintaining transmission in communities of persons with both low and high rates of partner change.
The Basic Reproductive Number
The basic reproductive number, R0, is defined as the average number of secondary infections generated by one infectious individual in an entirely susceptible population.1 In essence, it is a composite measure of transmission success in a defined community. For persistence, the magnitude of R0 must be greater than or equal to unity in value. Its magnitude determines the likelihood, speed, and scale of the spread of infection, encapsulating descriptions of the parameters that determine the typical course of infection (e.g., the incubation period and the typical duration of infectiousness) and those that determine transmission between hosts (e.g., rates of sexual partner acquisition). A simple formulation of the basic reproductive number for a sexual transmitted infection spreading in a homogeneously mixing population is as follows: EQUATION 1 where c denotes the average rate of sexual partner change, β records the transmission probability per partnership, and D is the mean duration of infectiousness. In this formulation, the size of the population has no impact on transmission success. For most directly transmitted childhood viral and bacterial respiratory tract infections, population density plays a key role in the rate of transmission. However, for STDs, population size or density will not in general influence the number of sex partnerships formed per unit time in a simple proportional manner. The concentration of many STDs in urban populations is thought to be a consequence of many different factors, including the influence of population density on sexual behavior and, hence, contact rates.11
Because of the limits placed on the number of potentially infectious contacts placed by the necessity for sexual intercourse occurring, STDs typically depend on relatively high transmissibilities per sexual contact and long periods of infectiousness in the infected individual to ensure long‐term endemic persistence.12 In developed countries, the short period between experiencing disease symptoms and seeking treatment for infections (e.g., gonorrhea) can severely restrict the magnitude of the basic reproductive number, such that persistence may depend on a small group of asymptomatic carriers.12 Asymptomatic infection is important for the transmission success of most sexually transmitted infections, including HIV, gonorrhea, and hepatitis B. The frequency of asymptomatic carriage may be related to gender. Even without effective chemotherapeutic treatment, abstinence from sex while symptomatic (promoted by public health education) can greatly reduce transmission success.13
The Simple Epidemic
Definition of the basic reproductive number and the stratification of the population into two classes (susceptibles and infecteds) under the assumption of homogenous sexual behavior provides a starting point for the exploration of many biological and behavioral complexities. It is illuminating, for example, to consider the pattern of a simple epidemic arising from the introduction of one infected person to a fully susceptible population. For illustrative purposes, we consider the example of HIV type 1 (HIV‐1) in a gay male or a intravenous drug‐using community. As depicted in Figure 1, a simple pair of differential equations denoting the rates of change over time in the density of susceptibles, X, and infecteds, Y, can generate epidemic curves that exhibit many similarities with observed patterns. Postintroduction, the incidence of infection increases to a peak before decreasing to a low endemic level, where reproductive success settles to a unity in value where each infection generates on average one new case. Concomitantly, the prevalence of infection rises in a sigmoid manner to a plateau that decays slightly over time because of disease‐induced mortality. One such observed pattern is plotted in Figure 2, which records the spread of HIV‐1 in intravenous drug users in New York City. A number of simple but important concepts are illustrated in these graphs.
Of most importance is the observation that incidence increases and decreases even in the absence of any intervention to slow spread. This observation is due to a saturation phenomena, where the magnitude of the effective reproductive number (the word effective implying that the population is not totally susceptible) decays as the rising number of infecteds reduces the likelihood that a new contact of an infected is with a susceptible person. It is clearly important to be aware of this effect to avoid drawing the possibly false conclusion that interventions have reduced the incidence of infection. Interventions will speed the decay in the bell‐shaped incidence curve from its peak or reduce the height of the peak, but the magnitude of their effect must be measured against the background influence of saturation.
At the endemic equilibrium state, the effective reproductive number equals one and is the product of the basic reproductive number, R0, and the proportion of the population susceptible (x*). In the absence of acquired immunity (which is rare for STDs that tend to induce persistent infections or infections that do not induce lasting immunity to reinfection, often due to antigenic heterogeneity in the pathogen population), the relationship between R0 and x* is simply x* = 1 ÷ R0. In reality, the endemic prevalences of many STDs are typically low. To explain why saturation occurs at a low prevalence, it is necessary to consider how heterogeneity in sexual behavior influences the definition of R0.
Heterogeneity in Sexual Behavior
Not everyone has the same risk of acquiring and passing on sexually transmitted infections to new partners due to heterogeneity in the sexual behaviors that facilitate transmission. If we assume that the mean rate of sexual partner acquisition is μ and the variance is σ2, then our definition of R0 becomes14 EQUATION 2
By inspection of this expression, it is clear that the magnitude of R0 is greatly influenced by the variance in sexual activity. This variance can be high, despite a low mean level of activity, if a small core of individuals change sexual partners frequently. In these circumstances, an infection could persist endemically despite low average rates of sexual activity. The proportion of the population remaining susceptible at equilibrium is no longer the inverse of the basic reproductive number. The lower the mean and the greater the variance in the rate of sex partner change, the more concentrated infection will be in the tail of the distribution of risk behavior.15 Typically, only a small fraction of the population will be infected, with the precise value depending on the summary statistics of the probability distribution of the rates of sexual‐partner acquisition. More generally, the simple models that encapsulated heterogeneity in the number of sexual partnerships formed per unit time helped to focus attention on the need to quantify these distributions in defined populations. Theoretical studies certainly played an important role in stimulating a number of large‐scale population‐based sexual behavior surveys in different countries that have provided much detail on these distributions stratified by variables such as gender and age.
Heterogeneity in sexual behavior is manifest in many of the variables that influence the likelihood of transmission. A further example is the distribution of the number of sex acts within a partnership. Assuming a simple binomial probability where infection during a sex act between an infectious and a susceptible person either occurs or does not occur, then the probability of transmission within a sexual partnership (β) is given by the relationship EQUATION 3 where γ is the probability of infection during a single act and a is the average number of sex acts per unit of time. If there is a low risk per act but many sex acts per partnership, this expression is essentially a Poisson process.16 Various modifications can be made to this expression, with different assumptions about the distribution of acts per partnership and the relationship between acts per partner and partners per unit time.17 For example, survey data in a variety of populations suggest that as the number of partners a person has per unit of time increases, the average number of acts with each partner decreases. In these circumstances, those with an intermediate number of partners concomitant with an intermediate number of sex acts with each partner can contribute most to the spread and persistence of an STD.18 This is most likely to be the case for an infection with a relatively low transmission probability per sex act, such as herpes simplex virus type 2.19,20
The prevailing relation between the distribution of sex acts and the value of the transmission probability is clearly important in any assessment of the potential impact of population‐based condom use as an STD‐control intervention. For example, simple models incorporating representations of the distribution of acts per partner suggest that condoms will have the greatest impact on net transmission within a community when used in partnerships that include few sex acts, or when an STD has a low transmission probability.
Many other forms of heterogeneity that are known to be important in transmission have been analyzed in various studies that use simple and complex mathematical models. These include concurrent partnerships, heterogeneity in infectiousness throughout the incubation period, variation at different spatial locations, and heterogeneity in the structure of the pathogen population (strain structure). A common feature of all such models is a growing degree of complexity in the definition of a basic reproductive number. As the number of stratifications of the host or pathogen population rises, the concept of a single overall reproductive number becomes increasing difficult to define. More generally, the definition of a number R0ij is made to represent the generation of secondary cases by an infected in group i and in susceptibles in group j. Here, the group stratification may be based on sexual behavior, spatial location, age, gender, or other sociodemographic variables. To illustrate this concept, we briefly consider one basic stratification by the rate of sexual partner acquisition and examine mixing between the different sexual activity classes. The central issue here is the question of “who mixes with whom” and the importance of this question to observed epidemiologic pattern.
Patterns of Sexual Mixing and Sexual Partner Networks
Understanding the relation between the pattern of sexual contacts within a population and the spread of infection has stimulated the development of new data‐collection methods,21,22 and methods for analyzing and interpreting observed patterns.23,24 In many respects, these methods rely on an old and well‐tried method of STD control; namely, contact tracing. Mathematical models have certainly played an important role in stimulating interest in the use of this approach as a research tool for assessing who mixes with whom. Patterns of sexual partner mixing23,25 and concurrent or overlapping sexual partnerships26,27 have emerged as important influences in the dialogue between mathematical modelers and those studying sexual behavior. It is clear from theoretical studies that the pattern of mixing between different sexual activity groups has a major influence on the rate and pattern of infection in a defined community. For example, if mixing is assortative (i.e., like with like) infection will be largely restricted to the small core of highly sexually active individuals with occasional transmission events in the lower‐activity groups. In some circumstances, the epidemic may develop as a series of waves of differing magnitudes as the infection spreads from one risk group to the next. The degree to which these waves are distinct will depend on the precise degree of assortative mixing. More generally, similar principles apply if we move up a scale to risk groups defined by habit or sexual predilection, such as intravenous drug users, homosexuals, and heterosexuals, or if we consider transmission between and within different spatial locations. If mixing is highly disassortative (i.e., like with unlike), the epidemic is likely to spread more slowly but affect a much higher fraction of the total community.
Despite a good general understanding of the impact of different mixing patterns on observed epidemiologic patterns, a number of important questions remain unresolved. For example, the recent construction of “individual”‐based simulation models has raised the issue of whether detailed network structures and spatial heterogeneity must be described if we are to fully understand the pattern of infection persistence in communities with active intervention programs. Preliminary work suggested that the most important determinant of STD prevalence is the partner choice of those in high‐activity groups, whereas persistence is dominated by the proportion of nonmonogamous partnerships that maintain chains of infection within populations.28 An improved understanding of the relationship between detailed patterns of sexual behavior and the incidence of different STDs is likely to play an important part in our thinking about how to best intervene to slow or eliminate spread.29
A number of recent modeling exercises have attempted to generate models that can be used by policy makers as tools that explore the impact of their policy. These “models” that provide numerical solutions tend to be named, and are provided with a “user‐friendly” interface, but can range widely in complexity from a deterministic model of the direct impact of interventions on HIV transmission events (AVERT)30 to a stochastic description of many concomitant STDs within a dynamic sexual partner network (STD‐SIM).31 Both of these model have been used to explore the action of observed interventions and carry on a tradition of modeling represented by the iwgAIDS model32 and Simul‐AIDS,33 where it is hoped that relevant complexity is captured by the available version of the model. These models provide a powerful means of communicating with policy makers. The challenge presented by the most recent generation of individual‐based simulation models is to translate the insights gained from specific simulations to general insights that are robust enough to be of use to policy makers.
The description of detailed sex‐partner network structures also poses a number of novel statistical questions in epidemiologic study. Empirical network data will necessarily be incomplete, and any method of sampling individuals provides a biased sample of the sex partnerships within a defined community.34 In exploring the resultant biases, individual‐based simulations of STD spread and different sampling methods allow the relation between model results and measures of network structure to be explored in a precise manner. The use of simulation models to explore the properties of statistical measures is an important development in the design and interpretation of field studies. Examples of the use of this general approach in STD epidemiology include the study of statistical measures of the influence of cofactor STDs on HIV‐1 transmission35 and the design of phase III HIV‐1 vaccine trials.36
Simple Insights and Policy Formulation
The growing volume of research that highlights the importance of various forms of heterogeneity to observed epidemiologic pattern argues for caution in the use of simple models that ignore much of this complexity to guide policy formulation. Simple models certainly help to understand what needs to be measured and how individual factors shape epidemic and endemic behavior, but can they be trusted to shape policy on interventions? The answer to this question is not straightforward. “All policy is based on theory, though not always on the best theory.”37 Whether policy based on verbal argument, experience, or intuition is better than that guided by simple theory deriving from mathematical models depends on many factors, not last of which is who's intuition and experience influences policy decisions. Although there are no general rules to govern when simple theory is a reliable guide and when it is not, a few simple examples serve to highlight what can be achieved. The pandemic of HIV and AIDS provides many examples, some of which have already been outlined previously; however, they deserve emphasis. Simple models served to highlight that a decline in incidence may not reflect the success of interventions but simply the natural course of the epidemic. Similar models highlighted that for long incubation‐period diseases, epidemic spread in low‐risk populations may take place over a period of many decades with a long period of almost imperceptible change, perhaps lasting for a decade or longer.14 This observation is of obvious relevance to the issue of whether there will be a major HIV‐1 epidemic in heterosexual populations in developed countries. These models also served as a template to highlight the many factors that should be quantified to fully understand the observed pattern of spread of HIV‐1 and heterogeneity therein, between and within different countries. These included changes in infectiousness over the incubation period of AIDS (based on viral load measures), distributions of sexual partner acquisition, probabilities of transmission, and patterns of sexual mixing.
The Demographic Impact of STDs
Assessing the impact of infectious disease on the growth and decay of human communities has typically been the preserve of historians and demographers. More recently, mathematical frameworks that meld epidemiology and demography have provided a new set of tools to evaluate the real and potential impact of given diseases. Bacterial STDs, such as chlamydia and gonorrhea, influence fertility by inducing tubal occlusion.38 Syphilis, and probably HIV, reduce fertility by increasing rates of spontaneous abortion.39 The potential impact of these mechanisms as proximate determinants of fertility has recently been explored using simple mathematical models with epidemiologic and demographic terms.38 The conclusions deriving from such studies simply highlight the great impact that these infections can have in poorer regions, where most STDs remain untreated.
The impact of HIV on fertility has only recently emerged as an issue of importance. By way of contrast, the impact of the virus on mortality and its demographic consequences have been a focus of debate for more than a decade. Early theoretical studies suggested that the full demographic impact of HIV would vary by location and would take decades to unfold.40 At the time, much was uncertain about the epidemiology and pathogenesis of HIV‐1, but the aim of early theoretical studies based on models that for the first time melded epidemiologic and demographic processes was to highlight the potential severity of the AIDS pandemic. The importance of the relation among prevalence of HIV, the incubation period of disease, and the population growth rate has been reiterated in subsequent mathematical modeling studies.41–43 Detailed data from Uganda based on longitudinal studies suggest that locally high HIV seroprevalences have dramatically reduced population growth rates by approximately 4% per capita per year. These dramatic reductions in population growth rates were observed in locations with high fertility and population growth before the emergence of HIV‐1. Such local impacts based on detailed studies are not necessarily representative of the picture at a national level.44 The HIV epidemic has been most severe in southern Africa,45 but its emergence in countries such as Zimbabwe and South Africa is also more recent so that its full demographic impact has not yet been felt. Nonetheless, mortality in Zimbabwe has started to climb rapidly,46 and across central and east Africa much of the gains in life expectancy achieved during this century have been dramatically over‐turned by HIV‐1 in the past decade.47
The Impact of Antivirals on HIV Transmission
The advent of combination drug therapy based on the inclusion of protease inhibitors for the treatment of early and advanced stage HIV‐1‐related disease has had a dramatic impact on AIDS‐induced mortality in the developed world in the last few years.48 Sadly, this trend is not repeated in the worst affected regions of the world, such as sub‐Saharan Africa, due to the current high cost of drug therapy. Therefore, the long‐term impact of novel drugs and treatment strategies is yet to be determined. Subcurative therapy that increases the life expectancy of those infected could have a perverse long‐term outcome if an increased average duration of infectiousness increases the incidence of new HIV infections.49 As such, the benefit of treatment to the individual may be at the expense of the community because of the enhanced opportunity for transmission over the lengthened incubation period. However, this somewhat gloomy prediction is unlikely to hold in practice, provided (as seems likely) the observed reductions in viremia induced by combination therapy are translated to reduced infectiousness. There are concerns about increased risk behavior in those receiving treatment. Concern is also targeted at the susceptible population where some individuals now falsely perceive HIV‐1 infection to be curable by chemotherapy. The major uncertainty, however, lies in the capability of the virus to evolve multiple drug resistance but retain its ability to spread by sexual contact. Multiply resistant strains have emerged, but their fitness relative to drug‐sensitive strains is unclear. An urgent need is to expand the mathematical model framework for the study of HIV‐1 to encompass the evolution of the virus under selection by community‐based combination therapy. A new generation of models are needed that meld epidemiology and population genetics within a multistrain framework. Such developments need to march hand‐in‐hand with the rapidly accumulating body of data about the frequency of resistance based on molecular epidemiologic studies.
The introduction of effective antivirals has provided a rich resource for the study of models of the population dynamics of HIV within the host.50,51 These models are similar to conventional epidemiologic frameworks, with the individual person as the host replaced by the CD4+ cell within the infected individual. Current models are simple in structure, but their predictions have been influential in furthering knowledge about HIV‐1 pathogenesis. They have provided a template for the estimation of viral and infected cell turnover rates, which are much higher than many envisaged. More development is needed in this area, both in statistical estimation procedures, the treatment of immunological responses within the models, and the description of pharmacokinetic and pharmacodynamic properties of specific drugs used alone or in combination. Detailed models of the interaction of the host's immune system with pathogen populations have much to offer in the study of the complex nonlinear interactions so typical of the human immune system.17 They have already provided new insights to the distribution of turnover rates of lymphocyte populations, pretreatment and posttreatment in patients with HIV infection.52 However, most experimental immunologists may still retain a strong allergy to mathematical models, despite recent successes in the HIV‐1 pathogenesis area.
The future for mathematical studies of biological systems seems bright, with a growing appreciation among biomathematicians of the need to confront prediction with observation, concomitant with a feeling within the experimental and biomedical research community that mathematics may have something to offer in a world that is becoming almost too rich in an ever‐growing mass of empirical information. In the STD field in particular, future research needs are many and varied. Of particular importance is the need to provide policy makers with robust tools to assess the relative benefits of different forms of intervention. If epidemiologic concepts can be grafted onto current health economic practices, frameworks could be developed for precise cost‐benefit analyses. Evolution is always central to the study of biological systems and in the STD epidemiology field, and theory needs to catch up with the burgeoning volume of molecular epidemiologic data regarding pathogen evolution and diversity. New mathematical templates are needed that meld genetics and epidemiology, taking due account of the different evolutionary pressures acting on the infectious agent within the host, and on the events leading to transmission between hosts. The study of evolution of drug resistance is a particular need, but others include the maintenance of strain structure and the impact of vaccines on antigenic diversity. Increased computational power will tempt many down the road of individual behavior models, and surely much will be learned from such simulation studies. Going from the particular to the general will, however, present many challenges that must be faced if such work is to guide policy formulation. Last, but by no means least, is the exciting challenge of using mathematical models to facilitate the statistical estimation of key epidemiologic parameters. Simulating transmission in heterogeneous populations provides an excellent template for the study of the performance of many widely used statistical measures of risk and association.
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