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Original Articles: Note

Evidence Undermining the Adequacy of the HIV Reproduction Number Formula

Potterat, John J. BA*; Muth, Stephen Q. BA*; Brody, Stuart PhD

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IN 1987, May and Anderson1 introduced what has since been the standard approach to model the population flow of HIV infection and traditional sexually transmissible diseases (STDs): the reproductive number formula, R = βcxD. R, or the reproduction number, represents the average number of newly infected contacts generated by infected persons. This number is 1.0 at equilibrium where, on average, cases replace themselves. A value less than 1.0 indicates declining incidence and a value more than 1.0 indicates epidemic spread. The reproduction number is obtained by multiplying the infection's transmission probability per sexual act or per sexual partnership (β) by the average number of contacts (c), the proportion of susceptibles (x), and the duration of infectivity (D). A major difficulty with deriving reliable reproduction numbers in the real world lies with obtaining good data for each parameter. Although data for parameters β, x, and D may be difficult to obtain empirically, they present no conceptual difficulties. However, obtaining trustworthy information to compute c is both conceptually and empirically challenging. We present theoretical and empiric evidence that this framework does not and cannot yield consistent results.

To illustrate the potential importance of structural elements in contact patterns (c), Klovdahl and colleagues2 present two graphs containing the same elements but differing in spatial conformation. Each graph consists of the same number of persons, the same number (and kinds) of sexual relationships, and the same configuration of sexual activity classes. It is shown that if infection were introduced into each network graph, its propagation would differ. In other words, although the value for c and its variance ratio would be the same in each network, each network population's reproductive number is likely to be different simply because of its spatial conformation.

At about the same time, several researchers began to model an important structural property of relationships: concurrency.3,4 In brief, graph-theoretic and stochastic models predict substantial amplification of disease transmission when risk partnerships overlap. Concurrent partnerships appear to be more efficient at boosting transmission than multiple partnerships that are conducted serially. Again, modeling partnership conformation (in this case, concurrency) is predictive of a different (in this case, higher) reproductive number than would be expected from simply computing the mean (and mean-to-ratio) number of partners for c.

Importation of the social network paradigm to the infectious disease field occurred in the early 1980s.5,6 During the late 1980s, investigators initiated prospective studies to delineate the influence of network structure on disease propagation. Results indicate that both macrostructural and microstructural elements are associated with enhancing or inhibiting STD-HIV transmission: (1) component size and distribution7,8; (2) location of infected persons within components, as measured by centrality indices7,8; (3) the distribution of lower-order or higher-order microstructures within components9,10; (4) the degree of interconnectivity of microstructures, an indicator of network cohesion9,11; (5) patterns of partner selection (assortative or disassortative)12,13; and (6) concurrency.14 The important network effects noted in these empiric studies were not accounted for by individual-level data, such as numbers or distribution of partners or risk behaviors. Importantly, these studies cast doubt on the ability of c to account for these conformational effects on observed STD-HIV transmission.

A recent investigation to measure reproductive numbers empirically provides compelling evidence for the inadequacy of c to represent transmission reality. To estimate reproduction numbers for genital chlamydial infection in a medium-sized community, we used a network approach to directly obtain from contact-tracing data the average number of secondary cases generated by index patients.14 The best predictor of a reproductive number exceeding unity was concurrency (overlapping partners), as opposed to serial monogamy, yielding an odds ratio of 3.2. Controlling for the interaction between concurrency and number of partners reduced the odds ratio for concurrency from 3.2 to 1.8, and the odds ratio for number of partners was reduced from 1.6 to 1.5. Such a result indicates that the structure of contacts (overlapping partners) amplifies the reproduction number at a rate greater than the sum of its parts (i.e., the result predicted by obtaining a value for c based on average number of partners and their variance). This direct empiric observation strongly supports the predictions of models that factor concurrency.3,4

Since the appearance of May and Anderson's original article, much evidence has accumulated substantiating the frequent lack of dependence between partner number and observed STD-HIV incidence and prevalence.15 The basis appears to be due only in part to the frequent invalidity of self-reported data about sensitive behavior, such as partner number. That this lack of association has been reported from studies in many parts of the world makes it especially persuasive.

Finally, Garnett and colleagues16 evaluated transmission parameters using data from the literature and from recently enrolled patients with gonorrhea. The investigators found that the standard model could only explain the persistence of gonorrhea in the community by using extreme values for β and D. They concluded that “the model fails to take into account the detailed sex-partner network structure.”

Recent theoretical and empiric evidence undermines the adequacy of the reproduction number formula that has commonly been used since the mid-1980s to predict the magnitude and direction of STD-HIV propagation. Its main weakness lies with how c, the contact pattern, is represented. As presently configured, not only can c yield inconsistent reproduction numbers, but the variable also fails to account for spatial features that are evidently strong determinants of reproduction numbers. Whether this shortcoming is remediable within the formula's framework (e.g., by reconceptualizing the computation of c) remains to be seen. Perhaps the present formula's applicability depends on scale. Its use may yield reliable results at the partial network level where, presumably, network conformation is less influential than at the more complete network level.

More direct ways of assessing conformational attributes of the ability of risk networks to sustain or inhibit STD-HIV transmission will provide a more reliable gauge of transmission potential than the formula R = βcxD. Efforts are currently underway to find this STD-HIV transmission model “Holy Grail.”17,18


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