with similar risk behavior tend to mix, or interact, primarily among themselves (biased mixing) rather than randomly with everyone (homogeneous mixing). The details of our risk-based, biased-mixing model form a logical, coherent framework for interpreting the currently available data for the United States, but before we launch into details we want to emphasize one critical insight.

Since the growth in number of AIDS cases is cubic, the doubling time for the epidemic (the time for the number of cases to double) is continuously increasing. By contrast, if the growth were exponential, the doubling time would remain constant. In the framework of standard epidemiological models, the observed lengthening of the doubling time for AIDS (and hence its decreasing relative growth rate) might be attributed to changes in people's sexual behavior as a result of learning about AIDS. That interpretation has been promulgated in the press and has fostered complacency about the efficacy of education. Unfortunately, it is false because the long incubation time from infection to AIDS means that the effects of learning could not have been seen in the data until very recently.

The people who developed AIDS in the early to mid 1980s were infected with the virus that causes AIDS (the human immunodeficiency virus, or HIV) in the late 1970s and early 1980s, long before learning could have affected a major fraction of the male-homosexual population. So behavior changes, if any, could not have been nearly enough to give cubic growth of AIDS in the late 1970s and early 1980s. Thus the impact of learning cannot explain the observed cubic growth. Another possibility to consider is that the combined effect (or convolution) of an exponential growth in HIV infections and a highly variable time for conversion from infection to AIDS yields a power law. After an initial transient, however, an exponential convoluted with any bounded conversion function is still an exponential, not a power law. Moreover, it is unlikely that the initial transient would have the long, clearly defined cubic behavior seen in the data.

We have looked with considerable diligence for possible causes of cubic growth other than behavioral changes due to learning. We have concluded that the riskbased, biased-mixing model presented here best fits the observations. Our model is an extension of an earlier risk-based model of May and Anderson. They assumed homogeneous rather than biased mixing of the susceptible population and so predicted an exponential for the early stages of the epidemic. We have drawn much from their work, but it was the contradiction between the theoretically nearly inevitable early exponential growth and the observed cubic growth that led us to the following biased-mixing model. We also realized that random mixing is sociologically unrealistic.

The general mathematical formalism for our model is presented in "Mathematical Formalism for the Risk-Based Model of AIDS." Numerical solutions for different assumptions about population mixing and variability of infectiousness are presented in "Numerical Results of the Risk-Based Model." Here we will present an intuitive and simplified version of the model that emphasizes the main features leading to cubic growth, the quantitative predictions of the model, and the questions about human behavior and HIV transmission that must be answered before we can determine whether the patterns we have identified for the past will continue in the future.

Cubic Growth of AIDS

The CDC data on cumulative number of AIDS cases in the United States between mid 1982 and early 1987 are shown in Fig. 1. Data for times prior to 1982.5 are not shown because they are statistically unreliable. Data collected since 1987.25 are also not shown because the surveillance definition of AIDS was changed in 1987. The effects of that change on reporting delays and/or on the cumulative number of AIDS cases have not been fully determined, but preliminary analysis suggests that

BREAKDOWN OF AIDS
CASES BY RACE

Fig. 3. Plots of the cube root of the cumulative number of AIDS cases among homosexual, blsexual, and heterosexual males and among females are all nearly straight lines, indicating that AIDS has grown cubically in each group. Further, extrapolation of the curves (dashed lines) indicates that cubic growth began in all groups at approximately the same time that it began in the population as a whole. (Intravenous drug users have not been removed from the subgroups and the heterosexual-male subgroup includes cases of transfusion-related AIDS among juvenile males.)

1/3

(Number of AIDS Cases)

30

20

20

10

30

1982

1984

1986

1988

Year

where Ao = 340 and A1 = 174.6. Thus, after an initial transient, AIDS cases grow as the cube of time.

Breakdowns of the data by sex or sexual preference (Fig. 3) and race (Fig. 4) again show the same form of cubic growth for each subgroup. This is surprising, since the fact that the sum of all the data is cubic requires that the separate cubics be synchronized in time, in this case to within less than six months. In addition to presenting our model for cubic growth, we will discuss a possible seeding process for the initial cases of AIDS (see "The Seeding Wave") that is consistent with both the assumptions of our model and the synchronization of cubic growth in various subgroups.

Currently (third quarter of 1988), the CDC reported a cumulative total of 74,904 AIDS cases under their expanded mid-1987 surveillance definition. Of those about 14 per cent fell under the new categories added in mid-1987. More difficult to determine are the effects of delays between diagnosis and reporting to the CDC caused by the redefinition. The median reporting delay prior to the redefinition was about 3 months, and adjustments made for those delays have visible effects 36 months into the past on a graph such as the graph shown in Fig. 1. After the redefinition in mid-1987, the median reporting delay lengthened to about two years, and the reporting situation is still in transition. Consequently, we must await further data before we can model the effects of the transient caused by the redefinition and determine whether or not cubic growth has continued to the present. Nevertheless, we can say with certainty that the growth in AIDS cases is still polynomial of degree less than 4.

Expected Exponential Growth

We start by showing that the initial growth of AIDS (or of any infectious disease) would be exponential provided the population was homogeneous and did not change its behavior. We assume AIDS is the long-term result of infection by HIV and derive an equation for the rate of growth in the number of infected persons. Let I be the number of persons infected at time t in a population of size N. Assume that a, the rate at which an infected person transmits the AIDS virus to others, does not vary with time nor from person to person. Then during the time interval dt the I infected persons in the population would infect al persons. But the fraction I/N of those al persons are already infected, and so the number of additional persons infected during dt is al – al (I/N); that is,

=

that is, the number infected grows as the mth power of time. Moreover, since the doubling time ta is inversely proportional to the relative growth rate m/t, ta increases proportionally to t. In particular, ta (2- 1)t. The growth of AIDS is cubic, so m 3 and tα = (√2 − 1)t ≈ 0.26t. The observed doubling time for the AIDS epidemic has increased linearly from less than 0.5 year to the current value of more than 2 years. That change in doubling time and relative growth rate (by more than a factor of 4) is dramatically different from the constant doubling time characteristic of exponential growth.

A Risk-Based Model

Any model for the spread of an infectious disease must take into account the mechanism of its transmission, the pattern of mixing among the population, and the infectiousness, or probability of transmission per contact. The primary mechanisms for transmitting the AIDS virus are sexual contact and sharing of intravenous needles among drug users. Since little is known about needle-sharing habits, we concentrate on transmission through sexual contact. Here we build on data from the homosexual and heterosexual community. The relative growth rate of infection a can be approximated as the product of three factors: the infectiousness i, or probability of infection per sexual contact with an infected person; the average number of sexual contacts per partner c; and the average number of new partners per time interval p. That is,

Each of the factors in Eq. 10 can be a complicated function. For example, data suggest that infectiousness i is, on average, between 0.01 and 0.001 and that it varies with time since infection and, perhaps, from individual to individual (more about that later). The new-partner rate and the average number of contacts per partner certainly vary among the population and may depend on age, place of residence, race, personal history, and more. The general model presented in "Mathematical Formalism for the Risk-Based Model of AIDS" allows for some of these variations, but here we pick out the simplest features that lead to cubic growth.

The first crucial assumption of the risk-based model is that the susceptible population is divided into groups according to level of engaging in behavior that can lead to infection. The risk behavior most often correlated with HIV infection in the male-homosexual population (as suggested by the early work of the CDC) is frequent change of sexual partner, which we quantify as new-partner rate. The other behavior we consider is frequency of sexual contact (which is equal to the product cp in Eq. 10). Both sexual contact and some new partners are necessary to cause the epidemic.