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.
1. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press, 1991.
2. Bernoulli D. Essai d'une novelle analyse de la mortalité causée par la petite vérole et des advantages de l'inoculation pour la prévenir. Mém Math Phys Acad R Sci Paris 1760:1–45.
3. Hamer WH. Epidemic disease in England. Lancet 1906; I:733–739.
4. Ross R. Some a priori pathometric equations. BMJ 1915; 1:546–547.
5. Cooke KL, Yorke JA. Some equations modelling growth processes and gonorrhea epidemics. Math Biosci 1973; 16:75–101.
6. Reynolds GH, Chan YK. A control model for gonorrhea. Bull Inst Int Stat 1975; 106:264–279.
7. Hethcote HW. Qualitative analyses of communicable disease models. Math Biosci 1976; 28:335–356.
8. Hethcote HW, Tudor DW. Integral equation models for endemic infectious diseases. J Math Biol 1980; 9:37–47
9. Yorke JA, Hethcote HW, Nold A. Dynamics and control of the transmission of gonorrhea. Sex Transm Dis 1978; 5:51–57.
10. Hethcote HW, Yorke JA. Gonorrhea Transmission Dynamics and Control. Lecture Notes in Biomathematics, 56. Berlin: Springer-Verlag, 1984.
11. Brunham RC. Core group theory: a central concept in STD epidemiology. Venerealology 1997; 10:34–39.
12. Brunham RC, Plummer FA. A general model of sexually transmitted disease epidemiology and its implications for control. Med Clin North Am 1990; 74:1339–1352.
13. Garnett GP, Mertz KJ, Finelli L, Levine WC, St. Louis ME. The transmission dynamics of gonorrhoea: modelling the reported behaviour of infected patients from Newark, New Jersey. Philos Trans R Soc Lond B Biol Sci 1999; 354:787–797.
14. Anderson, RM, Medley, GF, May RM, Johnson AM. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causitive agent of AIDS. IMA J Math Appl Med Biol 1986; 3: 229–263.
15. Anderson RM, May RM. Epidemiological parameters of HIV transmission. Nature 1988; 333:514–522.
16. May RM, Anderson RM. The transmission dynamics of human immunodeficiency virus (HIV). Philos Trans R Soc Lond B Biol Sci 1988; 321:565–607.
17. Anderson RM. The Croonian Lecture, 1994. Populations, infectious disease and immunity: a very nonlinear world. Philos Trans R Soc Lond B Biol Sci 1994; 346:457–505.
18. Garnett GP, Anderson RM. 1996 Sexually transmitted infections: insights from mathematical models. J Infect Dis 1996; 174 (Suppl 2):S150-S161.
19. Blower SM, Porco TC, Darby G. Predicting and preventing the emergence of antiviral drug resistance in HSV-2. Nat Med 1998; 4:673–678.
20. White PJ, Garnett GP. Use of antiviral treatment and prophylaxis is unlikely to have a major impact on the prevalence of herpes simplex type 2. Sex Transm Infect 1999; 75:49–54.
21. Woodhouse DE, Rothenberg RB, Potterat JJ, et al. Mapping a social network of heterosexuals at high risk for HIV infection. AIDS 1994; 8:1331–1336.
22. Laumann EO, Gagnon JH, Michael RT, Michaels S. The Social Organization of Sexuality: Sexual Practices in the United States. Chicago: Chicago University Press, 1994.
23. Gupta S, Anderson RM, May RM. Networks of sexual contacts: implications for the pattern of spread of HIV. AIDS 1989; 3:807–817.
24. Morris M. A log-linear modeling framework for selective mixing. Math Biosci 1991; 107:349–377.
25. Koopman J, Simon C, Jacquez J, Joseph J, Sattenspiel L, Park T. Sexual partner selectiveness effects on homosexual HIV transmission dynamics. J Acquir Immune Defic Syndr Hum Retrovirol 1988; 1:486–504.
26. Watts CH, May RM. The influence of concurrent partnerships on the dynamics of HIV/AIDS. Math Biosci 1992; 108:89–104.
27. Morris M, Kretschmar M. Concurrent partnerships and the spread of HIV. AIDS 1997; 11:651–658.
28. Ghani AC, Swinton J, Garnett GP. The role of sexual partnership networks in the epidemiology of gonorrhea. Sex Transm Dis 1997; 24:45–56.
29. Potterat JJ, Rothenberg RB, Muth SQ. Network structural dynamics and infectious disease propagation. Int J STD AIDS 1999; 10:182–185.
30. Rehle TM, Saidel TJ, Hassig SE, Bouey PD, Gaillard EM, Sokal DC. AVERT: a user-friendly model to estimate the impact of HIV/sexually transmitted disease prevention interventions on HIV transmission. AIDS 1998; 12(suppl 2):S27-S35
31. Korenromp EL, Van Vliet C, Grosskurth H, et al. Model-based evaluation of single-round mass treatment of sexually transmitted diseases for HIV control in a rural African population. AIDS 2000; 14:573–593.
32. Bernstein RS, Sokal DC, Seitz ST, Auvert B, Stover J, Naamara W. Simulating the control of a heterosexual HIV epidemic in a severely affected east African city. Interfaces 1998; 28:101–126.
33. Auvert B, Buonamico G, Lagarde E, Williams B. Sexual behavior, heterosexual transmission, and the spread of HIV in sub-Saharan Africa: a simulation study. Comput Biomed Res 2000; 33:84–96.
34. Ghani AC, Donnelly CA, Garnett GP. Sampling biases and missing data in explorations of sexual partner networks for the spread of sexually transmitted diseases. Stat Med 1998; 17:2079–2097.
35. Boily MC, Anderson RM. Human immunodeficiency virus transmission and the role of other sexually transmitted diseases: measures of association and study design. Sex Transm Dis 1996; 23:312–332.
36. Boily MC, Masse BR, Desai K, Alary M, Anderson RM. Some important issues in the planning of phase III HIV vaccine efficacy trials. Vaccine 1999; 17:989–1004.
37. Hobsbawn EJ. Industry and Empire. Harmondsworth: Penguin Books, 1969.
38. Brunham RC, Garnett GP, Swinton J, Anderson RM. Gonococcal infection and human fertility in sub-Saharan Africa. Proc R Soc Lond B Biol Sci 1992; 246:173–177.
39. Gray RH, Wawer MJ, Serwadda D, et al. Population-based study of fertility in women with HIV-1 infection in Uganda. Lancet 1998; 351:98–103.
40. Anderson RM, May RM, McLean AR. Possible demographic consequences of AIDS in developing countries. Nature 1988; 332:228–234.
41. Anderson RM, May RM, Boily M-C, Garnett GP, Rowley JT. The spread of HIV-1 in Africa: sexual contact patterns and the predicted demographic impact of AIDS. Nature 1991; 352:581–589.
42. Garnett GP, Anderson RM. No reason for complacency about the potential demographic impact of AIDS in Africa. Trans R Soc Hyg Trop Med 1993; 87(suppl)1:19–22.
43. Gregson SAJ, Garnett GP, Anderson RM. Is HIV-1 likely to become a leading cause of adult mortality in sub-Saharan Africa? J Acquir Immune Defic Syndr Hum Retrovirol 1994; 7:839–852.
44. Sewankambo NK, Wawer MJ, Gray RH, et al. Demographic impact of HIV infection in rural Rakai District, Uganda: results of a population-based cohort study. AIDS 1994; 8:1707–1713.
45. Tarantola D, Schwartlander B. HIV/AIDS epidemics in sub-Saharan Africa: dynamism, diversity and discrete declines? AIDS 1997; 11(suppl B):S5-S21.
46. Gregson S, Anderson RM, Ndlovu J, Zhuwau T, Chandiwana SK. Recent upturn in mortality in rural Zimbabwe: evidence for an early demographic impact of HIV-1 infection? AIDS 1997; 11:1269–1280.
47. United Nations Population Division. The Demographic Impact of HIV/AIDS: Report on the Technical Meeting. New York: United Nations, New United Nations, November 10, 1998.
48. Palella FJ, Delaney KM, Moorman AC, et al, the HIV Outpatient Study Investigators. Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection. N Engl J Med 1998; 338:853–860.
49. Anderson RM, Gupta S, May RM. Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1. Nature 1991; 350:356–359.
50. Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 1995; 373:123–126.
51. Wei X, Ghosh SK, Taylor ME, et al. Viral dynamics in human immunodeficiency virus type 1 infection. Nature 1995; 373:117–122.
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52. Ferguson NM, de Wolf F, Ghani AC, et al. Antigen driven CD4+ T-cell and HIV-1 dynamics: residual viral replication under HAART. Proc Natl Acad Sci U S A 1999; 96:15167–15172.