Changing our model to a 2-group study, model A yields (0.024, 0.005, and 0.091) optimized infectivities for primary, asymptomatic, and symptomatic stages, respectively, resulting in an underestimation of symptomatic stage infectivity by approximately two thirds. (This is in keeping with results in prior 2-group studies.) Two-group study model B results in optimized infectivities of (0.028, 0.0, and 0.089). Once again, the symptomatic stage is underestimated by approximately two thirds.
Similarly, if we collapse all 6 activity groups into a single average activity level, the symptomatic stage infectivity is underestimated (0.045, 0.000, and 0.088) in the best-optimized 1-group model. The goodness-of-fit error is 0.03 compared with 0.016 for our 6-group optimization fit. Even with high symptomatic stage infectivities, a 1-group model is unable to capture the beginning of the San Francisco epidemic before 1979 (see Fig. 11). Details about the comparisons between the different group optimized models are given in Table 3.
What these comparisons show is that using fewer than 6 activity groups increasingly makes infectivity estimates artifacts of the simplifying assumptions such as core/noncore or the single activity group used in prior modeling. The 6-group models reflect the diversity of the population and reflect the reported data. What they teach us is that late in the disease, when more than half of the population is infected, lower sexual activity groups are being infected but the epidemic is accelerating. The SFCCC data show that large numbers of persons become infected, but there are fewer contacts with susceptibles because those still susceptible have few contacts. Most of these infections result from partners in the symptomatic stage. This requires a higher symptomatic stage infectivity estimate than previous models, which failed to reflect the low activity of remaining susceptibles. After 1980, the gay epidemic becomes dominated by symptomatic stage transmission.
The data we use is from the San Francisco epidemic up to 1984. We have assumed that each individual has a sexual activity level that does not change significantly over the period we are modeling. The Bell-Weinberg study,39 a Kinsey Institute study of the 1969 San Francisco population, found that the most active 28% of men had 51 or more partners “in the past year.” The study also found that the men who were most active over their lifetimes (again, 28%) had 1000 or more partners (Table 4). Such a total requires many years of high activity levels, perhaps 1 or 2 decades or longer. Our model is appropriate for such a population. Of course, there are some individuals who change behavior, but a large fraction of the most active “core” population remains highly active when in the symptomatic stage. The epidemic went from 4.5% to 60% infected in just 5 years.
One can hypothesize populations with variable activity levels. Some earlier models assume that individuals vary in their activity level over time. For example, Koopman et al24 assume that this variation is quite rapid. They use 2 activity levels: a core with 5% of the population and the less active noncore. They assume that individuals remain in the core for an average of 1 year, stating, “Our models are not intended to reflect the transmission dynamics of any real population” (24, page 250). The assumption implies that there is virtually no correlation between the activity level of a man when he becomes infected and the activity level a few years later when he is in the symptomatic stage. (Under that assumption, the probability of an individual in the core remaining in the core for 6 years is e−6 or 1/400). Having people rapidly switch activity levels is quite similar to assuming that there is a single activity level.
Some may suggest, quite plausibly, that it is likely people in the symptomatic stage are less active because of the effects of HIV. If so, the symptomatic stage infectivity would have to be higher than our 0.299 to account for the large number of observed cases. This can be described by a mathematic relation. If you cut the number of contacts in half uniformly for the symptomatic stage and for all activity groups, the optimal symptomatic stage infectivity would be doubled so as not to decrease the number of new infections below the observed level.
To account for the large number of new infections in the latter stage of the epidemic, during which most of the susceptibles were from low-activity groups, one must have high symptomatic stage infectivity. Only the 6-group model is able to determine the high level of infectivity in the symptomatic stage.
What if activity levels declined as the epidemic progressed? It is likely that activity levels began to decline as people became aware of some new gay disease around 1984 or 1985.6 We have tried alternatives to our model, for example, by cutting the contact rate in half in 1983 through 1984. We then again determined the infectivities for the 3 stages that resulted in the best fit of the data. The main effect is that the symptomatic stage infectivity must be higher than in our standard model. The infectivities of the first 2 stages are largely determined by the need to fit the pre-1981 beginning of the epidemic when there were few symptomatic stage men. Decreasing the activity while maintaining the number infected results in higher symptomatic infectivity. Our main conclusion in this report is that the symptomatic stage is far more infectious than the earlier stages. Our conclusion remains valid when there is a decrease in sexual activity level as the epidemic progresses. Our symptomatic stage estimate, although higher than what prior studies report,6,16,24 is, in fact, a lower limit. Indeed, Hethcote and Van Ark's assumption of decreased activity beginning in 1981 requires them to use an AIDS stage infectivity of 0.75 to model the SFCCC data.14
Our model is a deterministic model. At each time step, we determine the fraction of individuals who make the transition from a given stage of infection to the next for each activity level, such as from core susceptible to core primary infection. We also developed a stochastic model in which, at each time step, we first compute the fraction of people who would make each transition according to the rules of our deterministic model and then convert this to a number of men by considering San Francisco gay population to be an estimated 70,000 individuals from the late 1970s through early 1980s.28 Using this fraction as a mean, we select a random number from a Poisson distribution. This random number becomes the number of individuals who make the transition at that time. The epidemic is then simulated, repeating this Poisson process for each time step and for every transition (see appendix). We find that the biggest differences between the epidemics of the deterministic model and the stochastic model occur when the fraction of people infected is small. By the time the fraction of infected individuals reaches 4.5% (as was the case in 1978, the SFCCC first data point), both models generate similar curves.
Because we are using the same 3 infectivities in both models, our confidence in these 3 infectivities increases. When their results are congruent, there is negligible harm in using a deterministic model, even when a stochastic model might seem more appropriate theoretically. Both approaches require high levels of symptomatic stage infectivity compared with primary stage infectivity.
We conclude that as compared with the assumptions used by other researchers, ours are more realistic and/or appropriate when applied to the SFCCC population. Our finding that symptomatic stage infectivities are approximately 30 times higher than previous estimates is not an artifact of our assumptions.
To test this conclusion further, we examine the cases in which the 2 infectivities are equal. For comparison, we ran our optimization code with the added constraint that the primary stage infectivity equals the symptomatic stage infectivity. The best-fit infectivities are 0.0, 0.02, and 0.0, respectively, for the 3 stages, and the RMS error is 0.036. Note that the RMS error is more than twice the RMS error for the best-fit infectivities mentioned previously. With this constraint, there is no best fit with positive values for the primary and symptomatic stage infectivities.
Allowing all combinations of the parameters mentioned previously, we end up with intervals of infectivity estimates. The primary stage varies from 0.014 to 0.024, the asymptomatic stage varies from 0.000 to 0.008,** and the symptomatic stage varies from 0.126 to 0.493. We are interested in the ratio of the symptomatic stage infectivity to the primary stage infectivity, which varies from 8.6 to 33.7. Thus, we conclude that even under variations of our model parameters, the symptomatic stage remains significantly more infectious than the primary stage.
Earlier models shed little light on slow transmission epidemics such as the African and San Francisco epidemics from 1980. They are dominated by symptomatic stage transmission, and they seriously distort the transmission dynamics after 1980. In 2-wave epidemics such as the San Francisco gay epidemic, there is a period when primary infection stage transmission is the predominant mode of transmission and alone can sustain the epidemic. Only symptomatic stage infection can sustain a slow epidemic such as the epidemic in Africa. If there is no such period, you get an epidemic such as that in South Africa. Ultimately, both epidemic patterns become dominated by symptomatic stage transmission.
Underestimating the symptomatic stage infectivity results in a severe underestimation of R0, the severity of the epidemic, and the measures necessary to end the epidemic. Our results provide a firm basis for a needed systematic reassessment of prevailing wisdom and strategies concerning the control, containment, and management of the HIV pandemic. Our results imply that screening of at-risk populations can identify most infected individuals before they enter their most infectious stage. Removal of symptomatic stage transmission would reduce R0 to less than 1 for many extant at-risk populations (although not the SFCCC population).
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For primary infection, the substages amount to the assumption that the average time after infection before one infects others is a 0.25 year. The peak of the viral load occurs at 0.25 year.10
We assume that susceptibles are in 3-month cohorts moving through susceptible and HIV infection stages as determined by the following variables and equations, where i, j = activity groups 1 (core) and 2 through 6; k = primary, asymptomatic, symptomatic; and t = 1, 2, 3, …. We assume that primary infection lasts 2 time periods but that all primary stage transmission occurs at the end of the first time period (ie, 0.25 year). We assume that new susceptibles enter the population at the same rate at which there are AIDS deaths.
We now define the variables used in our model:
- Susceptible fraction of group i at time t:
- Primary first fraction of group i at time t:
- Primary second fraction of group i at time t:
- Asymptomatic first fraction of group i at time t:
- Asymptomatic second fraction of group i at time t:
- Asymptomatic third fraction of group i at time t:
- Symptomatic fraction of group i at time t:
- Death fraction of group i at time t:
- Rate of partnering of group i with group j: rij
- Infectivity of persons at stage k: ak
- Fraction of group j that is in stage k at time t:
- Duration of asymptomatic substages: d
- Time step (in terms of fraction of a year): Δt
- From this model we can derive:
It is well established that in the first approximation, a typical untreated HIV infection progresses through the stages defined by successively high, low, and then very high viral levels in the blood.6 For the SFCCC study, the average durations of each stage are well documented and confirmed by modeling.6 The problem is that those averages are based on some people who pass through stages to AIDS faster and others who pass through slower or perhaps never become symptomatic or develop AIDS. This is only an issue for the relatively long asymptomatic and symptomatic AIDS stages. A good model needs to have some sort of diffusion pattern producing different rates of passage through those stages. Because there are no data regarding such “diffusion,” the simplest way to model this is to allow substages, where the number of substages determines the diffusion gradient. One can then choose how many substages to use depending on accuracy of best fit to HIV incidence with different numbers of substages. Our basic model does this and has 3 diffusion substages for the asymptomatic period and 1 stage for symptomatic/AIDS (see Fig. 5).
Most model parameter values are specified using SFCCC data. Average time from seroconversion (development of antibodies) to death was reportedly 10.3 years.†† In our model, the average primary infectious period lasts for 0.5 year, although seroconversion typically occurs at approximately 3 months. When we tried shorter primary stage periods such as 0.33 year, we were unable to fit the data as well. The epidemic behaves as if semen remains infectious for a bit longer than the usual primary stage period. Thus, the average duration of an HIV infection is 10.5 years. The remaining parameters are determined by best-fit approximation to the 1978 through 1984 SFCCC HIV infection growth data. Thus, every parameter value in the stage model is a firm SFCCC datum or is highly constrained by the SFCCC data. No unconstrained parameter assumptions are used.
Interpreting the Stage Model
The San Francisco population is divided into 6 activity groups, and each group is divided into 3 stages of infection (primary, asymptomatic, and symptomatic). To run our model, the user specifies the fraction of the men in each stage at an initial time, t0, and the 3 infectivities for the 3 stages of infection.
Given the fraction of each activity group that is in each stage at time t, the rules built into the model dictate what the corresponding fractions will be at time t + Δt, where Δt is a specified fraction of a year. We typically took Δt to be 0.25 or 0.33 year for the time step, and we report results here for 0.25 year, although the results for 0.33 year are similar. The model takes these fractions and takes another time step, applying the same procedure to obtain the corresponding fractions for time t + 2 Δt. The model takes a certain specified number of steps long enough for it to create a record of an outbreak similar to San Francisco's.
Model's Bookkeeping of New Infections in 1 Time Step
Given the fraction in each stage of each activity group, the model computes the expected number of contacts for all the men in each of the 4 stages: Nsus, Np, Na, and Ns. The risk, R, of a susceptible man becoming infected from 1 contact is:
In each activity group, the fraction of men newly infected, F_new, at time t + Δt is the fraction susceptible times the number of contacts each man has in time Δt times the risk, R. To obtain the susceptible population for each activity group for the time t + Δt, we subtract the fraction F_new from the susceptible fraction for time t and add F_new to the primary stage for time t + Δt.
Model's Bookkeeping of the Fraction in Each Stage
If the average duration for a stage is Y years, the fraction Δt/Y of people in that stage is moved to the next stage. We use 2Δt years for the duration of primary stage, 7 years for the duration of asymptomatic stage, and 3 years for the duration of symptomatic stage.
It should be noted that the real meaning of the 2Δt year primary period (which seems rather long to us) is that people who are initially infected at time t can create new infections at time t + Δt and the number of contacts they have while in the primary stage is 2Δt times the number of contacts for a year. One could alternatively say that the primary period is Δt and double the infectivity for the period.
Initializing the Model
We do not know when the epidemic actually began in San Francisco nor do we know the initial state, which really does not matter. No matter how we initialize the outbreak (whether the initial man or men were highly active or less active or in the primary stage or the symptomatic stage), we must choose an initial time so that the epidemic reaches the prevalence (ie, the fraction infected) of 4.5% in 1978. By that time and thereafter, the distribution of infected people is essentially independent of how we started the epidemic. As long as the initial infected fraction is small, the long-term shape of the plot of prevalence is not affected. The first prevalence report from the SFCCC study was 4.5% in 1978.
By our estimate, the 4.5% prevalence figure might better be reported as 4.5% ± 1.3%, representing an error of 1 standard deviation. The method for computing prevalence for other years is reported in less detail, and we cannot compute standard deviations for those years.
Optimization considerations determined the choice of 3 equal-duration asymptomatic substages. There are 6 activity level groups (see Table 2). Fractions of susceptibles who become infected in a given period are estimated using products of stage infectivities, fractions of groups in a given stage, and partnering rates between groups. Passage from a stage or substage to the next is proportional to average stage or substage duration. Using SFCCC data for other model parameters, stage infectivities are estimated by optimized fits to SFCCC HIV incidence data (see Table 1). An optimization routine (Newton's method applied to the gradients of E = root-mean-square errors in SFCCC fit to data for the years 1978 to 1984) was used to obtain best-fit approximation to SFCCC data estimates for average primary, asymptomatic, and symptomatic stage infectivities. This produced the solution in Figure 6.
Although the SFCCC HIV incidence data are of unusually high resolution and are biologically based, they are still subject to measurement and sampling errors. Furthermore, the behavioral data are self-reported. Studies indicate that such self-reported behaviors among gay men are quite reliable.41-43 Nevertheless, there are errors inherent in such data.
When we vary the fraction of the population infected for the data points 1978 through 1984 by approximately ±0.02 and then apply these optimization procedures, we get slightly different infectivities for our 3 stages. In summary, primary stage infectivity varies in the range of 2.3% to 2.5%, asymptomatic stage infectivity varies in the range of 0.0% to 0.4%, and symptomatic/AIDS stage infectivity varies in the range of 25.4% to 34.4%. Figure 12 shows us the region of uncertainty.
Fraction of Infecteds in the Bathhouse by Activity Group
From the model equations, we calculate the proportion of infecteds in each activity group versus time as follows:
The results are shown in Figure 11.
As mentioned previously, we developed a stochastic model in which, at each time step, we first compute the fraction of people who would make each transition according to the rules of our deterministic model and then convert this to a number of men by considering the San Francisco gay population in the late 1970s and early 1980s to be an estimated 70,000 individuals.28 Using this fraction as a mean, we select a random number from a Poisson distribution. This random number becomes the number of individuals who make the transition at that time. The epidemic is then simulated repeating this Poisson process for each time step and for every transition.
Let fi be the fraction of population in activity group I, and T be the total population (N = 70,000).
- Stochastic model
2. Repeat procedure
- (fraction that transitions from P1 to P2 at time t)
- 1.2. Let
- 1.3. Compute random variable from 0 to 1, r.
- 1.4. Find N such that:
- 1.5. Let:
*If the RAI is for only 1 partner, we are overestimating the number of contacts by a factor of 2, resulting in a 50% underestimation of stage infectivities, but not affecting the ratios of stage infectivities. Thus, we divide the population into 6 groups based on this estimated average RAI activity (contacts). The average number of RAI partners per year is 48.6,23
†We can understand much of the early dynamics without finding a solution to the model. Of course, any active man could have become infected and then infected others, but it is instructive to focus on the core group and only the infections the members of the core group caused when in the primary stage. Those men had 231 partners per year or approximately 115 in the primary stage (0.5 year). Forty-eight percent of those contacts (n = 55) are with men in the core group. The primary stage infectivity is 2.4%, resulting in 1.3 infections. Some of these infections are in the first quarter of the year, and some are in the second quarter; the average time to infection is 0.25 year. Hence, each quarter of a year, the number of infected men grows by a factor of 1.3. The result of 4 such steps (1 year's worth) is a growth with a factor of 3. We see then that the core primary stage men in San Francisco were able to drive the first (fast) wave of the epidemic.
‡Studies of heterosexual populations show that untreated infected individual's viral loads follow a pattern of moderate, then low, and then high as a person progresses through the disease.29-32 This corresponds to the pattern of our infectivity estimates.
§Our model has a time step of 0.25 year, 2 substages for the primary period, 3 substages for the asymptomatic period, and 1 substage for the symptomatic period. We also considered time steps of 0.33 and 0.5 year, along with 1 to 2 substages for the primary stage, 1 to 6 substages for the asymptomatic stage, and 1 to 2 substages for the symptomatic stage.
**If an infectivity value becomes negative in our optimization routine, we set it to 0. We do not believe that the asymptomatic period is ever truly 0.
††Individuals undergoing antiretroviral treatment tend to progress to AIDS at a much slower rate than untreated individuals.4