In 2001, Friedman et al. conjectured the existence of a “firewall effect” in which individuals who are infected with HIV, but remain in a state of low infectiousness, serve to prevent the virus from spreading. To evaluate this historical conjecture, we develop a new graph-theoretic measure that quantifies the extent to which Friedman’s firewall hypothesis (FH) holds in a risk network. We compute this new measure across simulated trajectories of a stochastic discrete dynamical system that models a social network of 25,000 individuals engaging in risk acts over a period of 15 years. The model’s parameters are based on analyses of data collected in prior studies of the real-world risk networks of people who inject drugs (PWID) in New York City. Analysis of system trajectories reveals the structural mechanisms by which individuals with mature HIV infections tend to partition the network into homogeneous clusters (with respect to infection status) and how uninfected clusters remain relatively stable (with respect to infection status) over long stretches of time. We confirm the spontaneous emergence of network firewalls in the system and reveal their structural role in the nonspreading of HIV.
Social network research among people who inject drugs (PWID) has produced considerable data on HIV-1 infection profiles and equally detailed data on the broad demographic and behavioral profiles of injecting communities and their risk behaviors. However, prior research has not—and for reasons of cost often cannot—produce long-term, dynamic data on these same populations. Risk networks—graphs whose vertices are individuals and edges are social connections bearing disease transmission risk—are now widely recognized as a critical construct in understanding infection patterns [
HIV has been investigated extensively in a number of PWID communities, including New York City [
The mathematical model underlying the stochastic discrete dynamical system consists of three parts: (i) the
Within a PWID risk network, each node is an individual and each edge represents a relationship that bears the potential for injection drug couse—referred to hereafter as
In what follows, we adhere to the standard mathematical conventions: given a set
(A)
In the context of this work, we drew upon data collected in the Social Factors and HIV Risk study (SFHR). Conducted between 1990 and 1993, SFHR was a cross-sectional, mixed methods project that asked 767 out-of-treatment intravenous drug users about their risk networks and HIV risk behaviors in the prior 30 days. Interested in both individuals' network composition (namely, the presence of high-risk partners) and sociometric risk position, the SFHR study produced several major findings relevant to risk populations with high HIV prevalence and low secondary incidence [
(B)
Given a risk network
In the next section, we present the statistical network model extracted from the SFHR risk network.
(C)
Significant Attributes (as determined by ERGM).
Name | Possible values ( |
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|
{Male, Female} |
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{White, Hispanic, African-American, Other |
|
{[15–20), [20–25), [25–30), [30–35), [35–40), [40–45), [45–50), [50–55)} |
|
{[0–2), [2–4), [4–10), [10–20)} |
Gender univariate
Male | Female | |
---|---|---|
|
541/767 | 226/767 |
Ethnicity univariate
White | Hispanic | African-American | Other | |
---|---|---|---|---|
|
243/767 | 206/767 | 311/767 | 7/767 |
AgeBinned univariate
[15–20) | [20–25) | [25–30) | [30–35) | [35–40) | [40–45) | [45–50) | [50–55) | |
---|---|---|---|---|---|---|---|---|
|
6/767 | 32/767 | 158/767 | 172/767 | 198/767 | 159/767 | 23/767 | 19/767 |
DegreeBinned univariate
[0–2) | [2–4) | [4–10) | [10–20) | |
---|---|---|---|---|
|
322/767 | 221/767 | 161/767 | 63/767 |
Gender bivariate
|
Male | Female |
---|---|---|
Male | 556/1032 | 180/1032 |
Female | 180/1032 | 116/1032 |
Ethnicity bivariate
|
White | Hispanic | African-American | Other |
---|---|---|---|---|
White | 232/1032 | 27/1032 | 73/1032 | 4/1032 |
Hispanic | 27/1032 | 222/1032 | 57/1032 | 7/1032 |
African-Am. | 73/1032 | 57/1032 | 238/1032 | 21032 |
Other | 4/1032 | 7/1032 | 2/1032 | 0/1032 |
AgeBinned bivariate
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|
|
2/1032 | 1/1032 | 3/1032 | 0/1032 | 0/1032 | 0/1032 | 0/1032 | 0/1032 |
|
1/1032 | 2/1032 | 12/1032 | 8/1032 | 8/1032 | 5/1032 | 1/1032 | 0/1032 |
|
3/1032 | 12/1032 | 54/1032 | 63/1032 | 48/1032 | 21/1032 | 3/1032 | 0/1032 |
|
0/1032 | 8/1032 | 63/1032 | 76/1032 | 71/1032 | 37/1032 | 7/1032 | 2/1032 |
|
0/1032 | 8/1032 | 48/1032 | 71/1032 | 88/1032 | 55/1032 | 10/1032 | 8/1032 |
|
0/1032 | 5/1032 | 21/1032 | 37/1032 | 55/1032 | 56/1032 | 4/1032 | 8/1032 |
|
0/1032 | 1/1032 | 3/1032 | 7/1032 | 10/1032 | 4/1032 | 2/1032 | 0/1032 |
|
0/1032 | 0/1032 | 0/1032 | 2/1032 | 8/1032 | 8/1032 | 0/1032 | 2/1032 |
DegreeBinned bivariate
|
|
|
|
|
---|---|---|---|---|
|
134/1032 | 74/1032 | 41/1032 | 20/1032 |
|
74/1032 | 180/1032 | 96/1032 | 49/1032 |
|
41/1032 | 96/1032 | 60/1032 | 32/1032 |
|
20/1032 | 49/1032 | 32/1032 | 34/1032 |
(D)
Each execution of AddEdge takes place in the context of a specific vertex
(
To understand the three biases used in defining
Individual agency may drive PWIDs to leave the risk network over time. To model this, each node
Whenever the network lifetime
The new individual
Each individual
In the context of this work,
During each risk act, the likelihood of viral transmission is 0 if both individuals have the same infection status. If the individuals are serodiscordant (i.e., precisely one of them is infected), then the probability of transmission is modeled by an
A two-parameter representation of HIV infectiousness as a function of infection age.
In the context of this work,
At a time Type 1. The risk network Type 2. An HIV− individual whose viral burden is in the chronic low-infectiousness phase cannot be reinfected through new risk behaviors, and so cannot return to a state of acute infectiousness. When such an individual separates HIV− nodes from acute HIV+ individuals, it obstructs the transmission of the virus from the latter to the former.
In this section, we formally define a graph-theoretic measure which captures the extent to which HIV− individuals can attribute their uninfected status to the two types of network obstructions described above.
Towards this, let
The top image in Figure
(Top) Risk network with 10 HIV+ (4 acute) out of 26 nodes; (bottom) virus-centric view, FW:
Now consider the human-centric view of a topologically isomorphic risk network shown at the top of Figure
(Top) Risk network with 10 HIV+ (4 acute) out of 26 nodes; (bottom) virus-centric view, FW:
In this section, we use the SFHR-based statistical network model (with a modified pathogen prevalence parameter
HIV rates in PWID networks of size 1 k–25 k nodes.
To test the firewall hypothesis, the system was frozen in mid-trajectory at monthly time intervals so the value of the FW measure could be computed (as defined in (
Firewall Hypothesis Validity in PWID networks of size 1 k–25 k nodes.
Number of acute HIV infections in PWID networks of size 1 k–25 k nodes.
To facilitate further analysis, we divided the trajectories, referring to the first 18 months as the emergent period, and the 13+ later years as the steady period. Each of the periods is treated in turn in the sections that follow.
To understand the behavior of the FW measure along system trajectories, we return to its definition as the quotient of the number of firewalled individuals by the number of HIV− individuals. The number of firewalled individuals in a risk network of 25,000 nodes is depicted over the 18-month emergent period in the left graph in Figure
The emergent period: (a) firewalled nodes; (b) HIV prevalence, number of HIV− individuals.
The firewall effect during the emergent period.
(a)
(b)
As in our consideration of the emergent phase, the graphs discussed here are drawn from 10 trials of 25,000-node PWID networks drawn from the statistical network model extracted from the SFHR data set. We now consider the dynamics of the FW measure's numerator and denominator during the steady period, after the hot spike in new infections has subsided. During the steady period, the number of firewalled individuals (numerator) is seen to decline from 12,000 to 10,000 over the 13+ years of the simulation (see Figure
The firewall effect during the steady period.
(a)
(b)
Having conducted simulation experiments based on the SFHR model and used these to demonstrate the occurrence of subsaturation stabilization via the firewall hypothesis, we acknowledge that the model contains a large number of parameters. While these parameters were set to consensus estimates derived from the ethnographic data collected as part of the SFHR study, it would be natural to ask whether the parameter settings had a significant impact on the emergence of subsaturation stabilization and the firewall phenomena in the above experiments. Significant model parameters included the following: from Section From Section From Section
In summary, our conclusions concerning the emergence of subsaturation stabilization and the firewall phenomena (which were drawn from simulations parameterized by data from the SFHR study), are in fact robust within a wider range of model parameter settings, though the extent of the two phenomena (as evaluated by stabilized HIV prevalence levels and FW measure values) is certainly nominally influenced by the specific choice of model parameter settings.
Having described a stochastic dynamical system modeling a dynamic PWID risk network, whose simulated trajectories match the historical HIV dynamics known for PWID networks in New York City during the early 1980s, we determine that nodes with mature HIV+ status tend to divide the network into clusters of uninfected nodes that remained relatively stable over time. Thus, the FH holds significantly (for up to 80% of uninfected individuals) and so captures an important barrier to HIV propagation in PWID risk networks. In considering the microlevel mechanics underlying the emergence of the FH, we find it helpful to examine the network during two phases of HIV infection: an initial, emergent phase of rapid spreading and a later period of stable HIV rates. There is an enduring presence of new infections that fail to propagate during the stable phase, and this is because of the structural effects created during the emergent phase when a significant fraction of uninfected nodes coalesce into small components (in the virus-centric view of the network). These small clusters represent margins of the network and are often composed of a few (or even single) individuals. Small components ensure that the ability for new infections to spread
Our research also suggests that overall network size plays a key role in HIV dynamics among injecting drug user networks. Consistently, networks of size 5,000 through 25,000 behaved within a narrow (and therefore predictable) range of overall characteristics. Networks of 1000 nodes or fewer, on the other hand, showed high variability in their network-wide behavior. This latter finding bears serious consideration for those concerned with interventions aimed at influencing the overall rate of HIV among injecting drug user networks. If smaller networks show high variability in their dynamics—leading to the idea that they are more subject to stochastic events than networks of large size—then understanding where and how particular interventions will succeed or fail becomes very difficult. What such variability in outcomes indicates is that stochastic factors may outweigh node level dynamics in determining network-wide outcomes through time in small networks. Put another way, the outcome of interventions in small-scale networks may not serve as good indicators of likely outcomes of the same intervention in other small networks, nor in the same networks at a different time, nor in large networks. In each case, the effects of random events may render the otherwise most successful interventions moot, or the most ill-adapted interventions successful—this without a change in the underlying set of network attributes or dynamics. We recognize that this finding represents a difficult challenge to policies advocating demonstrated evidence-based interventions (DEBIs) [
Simulation of formal dynamical systems is far from demonstration of actual disease dynamics, of course. But the results of this project can point to ways that network wide phenomena are shaped by local social processes, and thereby open avenues for future research that may be hidden by the limits of more standard empirical investigation. We note that the disease dynamics reflected here may be partly explained by the social circumstances that produce the SFHR PWID risk network (on which the simulation topologies were based). Among the most important of these is the central role that shooting galleries played as venues for drug use at the time of the SFHR study [
These circumstances may have, at least initially, helped promote the firewall effect described here. Under such conditions, new users with few network connections are likely to find themselves in shooting galleries with shooting gallery operators who were both of high degree and more likely to be in a state of mature infection, and thus effective firewalls against new infections potentially moving through the network. Conversely, as police interdiction gradually came to target shooting galleries (and shooting gallery operators became targets of police arrest), the disruption of stable relationships and the removal of critical central nodes from the network may disrupt this firewall effect, forcing remaining network members to seek out new sources and injection partners. This would have the effect of significantly reorganizing the network (in the virus-centered view). Police decisions were obviously weighted by other concerns, but an important suggestion of the simulation results presented here is that the public health implications of those decisions are likely difficult to gauge. Drug interdiction strategies are seldom seen as increasing risk, and it would likely seem highly counterintuitive that the removal of HIV+ individuals who have been infected for more than three months and who play a brokerage role in the network may in fact raise the level of risk for the remaining risk network—but that is what is suggested here. Such conclusions are obviously highly speculative. But they come as the results of models and simulations whose scale and scope cannot be matched by more direct empirical research. It remains before us to translate these suggestions into concrete research strategies capable of testing and evaluating both these results and their implications for public policy.
The authors would like to thank the referees for their many helpful suggestions and comments which have strengthened the exposition of these results. This project was supported by NIH/NIDA Challenge Grant 1RC1DA028476-01/02 awarded to B. Khan/K. Dombrowski, the CUNY Research Foundation, and John Jay College, CUNY. The authors also gratefully acknowledge many insightful discussions with the researchers at the Center for Drug Use and HIV Research (NYU College of Nursing) in the course of this work with support from Grants P30DA11041 (Center for Drug Use and HIV Research) and R01 DA006723 (Social Factors and HIV Risk). The funders had no role in or responsibility for the study design, data collection and analysis, conclusions, decision to publish, or preparation of the paper. The analyses discussed in this paper were carried out at the labs of the New York City Social Networks Research Group (