Size matters: concurrency and the epidemic potential of HIV in small networks

Abstract

Background: Generalized heterosexual epidemics are responsible for the largest share of the global burden of HIV. These occur in populations that do not have high rates of partner acquisition, and research suggests that a pattern of fewer, but concurrent, partnerships may be the mechanism that provides the connectivity necessary for sustained transmission. We examine how network size affects the impact of concurrency on network connectivity.

Methodology/principal findings: We use a stochastic network model to generate a sample of networks, varying the size of the network and the level of concurrency, and compare the largest components for each scenario to the asymptotic expected values. While the threshold for the growth of a giant component does not change, the transition is more gradual in the smaller networks. As a result, low levels of concurrency generate more connectivity in small networks.

Conclusions/significance: Generalized HIV epidemics are by definition those that spread to a larger fraction of the population, but the mechanism may rely in part on the dynamics of transmission in a set of linked small networks. Examples include rural populations in sub-Saharan Africa and segregated minority populations in the US, where the effective size of the sexual network may well be in the hundreds, rather than thousands. Connectivity emerges at lower levels of concurrency in smaller networks, but these networks can still be disconnected with small changes in behavior. Concurrency remains a strategic target for HIV combination prevention programs in this context.

Figure 1. Growth of the largest component in large networks at the threshold: This figure demonstrates the rapid growth of the giant component in large networks near the threshold level of concurrency.

The top row shows histograms of the momentary degree distribution for networks at mean degree 1.68, 1.74, 1.80 and 1.86, corresponding to 56, 60, 64 and 68 percent of sexually active persons having concurrent partners. The bottom row gives a visualization of a typical largest component. We can see that the percentage of the population in the largest component jumps rapidly from 2% to 64% with a very small increase in mean degree (0.2 of a partner, 12% more persons with concurrent partnerships).

Figure 2. Mean size of the largest component as a function of network size and concurrency: We see here the average size of the largest component for networks of size 100, 200, and 500 nodes as the concurrency rises through the threshold level.

The black line gives the analytic approximation for large networks. Notice that the curves for the small networks approach the analytic result as the network size increases. The small networks have the same threshold, but the behavior around the threshold is different. The flatter curve means that lower levels of concurrency produce higher levels of connectivity in small networks.

Figure 3. Distributions of the largest component size and density as a function of network size and concurrency: In this figure we reproduce some of the information in Figure 1 for networks of size 100, 200 and 500.

The top row again shows histograms of the momentary degree distribution for networks at mean degrees around 1.4, 1.6 and 1.8, corresponding to concurrency prevalence of 35, 50 and 65%. The middle row shows the size distribution of the largest component as a percentage of the network, and the bottom row shows the distribution of the density of the largest components. Note that the percentage of the network in the largest component decreases with network size, but variability rises with concurrency levels. The density of the largest component decreases with network size and level of concurrency.