Editorial Commentary: Network Epidemic Models: Assumptions and Interpretations

(See the HIV/AIDS Major Article by Palombi et al, on pages 268–75.)

Palombi et al [1], address several shortcomings of the article by Granich et al [2], which investigated the potential impact on human immunodeficiency virus (HIV) epidemic dynamics of a policy to treat all HIV-infected patients. Palombi et al make use of a subject-level model to incorporate in their simulation study transmission rates that were functions of duration of infection, adherence levels, mortality, and retention rates. Their model also allows for the incorporation of network features, such as concurrency (the occurrence of simultaneous sexual partnerships) or degree (the number of current sexual partners). Below, we describe some of the assumptions that underlie these models and consider the impact of different choices of input parameters; our analysis suggests caution in interpreting results.

Palombi et al incorporated concurrency by using a network model developed by Kretzschmar and Morris—hereafter referred to as KM [3]. Epidemic models have shown that concurrency affects the spread of sexually transmitted infections [4]. Because the network model governs the evolution of the epidemic model used by Palombi et al, it is essential to consider this model carefully.

The KM model has 4 key components: probability of pair formation, probability of pair separation, a rule for partner mixing, and level of concurrency in the population, quantified using an index, κ [3]. It is not straightforward to understand the nature or the implications of the assumptions of the KM model, so we carried out a simulation study using the model to obtain further insight. We were not able to evaluate the exact settings of Palombi et al, because a complete set of the parameter values for their model are unavailable. For example, Palombi et al state that their sexual mixing was disassortative (high-degree subjects prefer partners with a low degree and vice versa) without providing information about the degree of disassortativity. The rationale for this assumption was also not clear. Furthermore, given 3 of the 4 components (mixing patterns, concurrency, and the probabilities of pair formation and separation), the fourth component is fixed. Preferential partner mixing is associated with higher levels of concurrency when the probabilities of pair formation and separation are fixed [5]. It would be useful to specify which parameters are estimated from data and which are forced.

In the absence of information about the values in the study by Palombi et al, we set the probabilities of pair formation and separation (.01 and .005 per day, respectively) to values that were also used by KM. We considered both random partner mixing (a subject's current degree does not cause him or her to prefer a new partner based on his or her degree) and disassortative mixing, as assumed by Palombi et al. The mixing function, which is the probability of forming a relationship between 2 individuals depending on their degrees, is a function of a tuning parameter, ζ, which allows for a continuous transition between mixing patterns and modifies the level of concurrency κ [3].

One aspect that we explored was the implications of the KM model for the fraction of persons that have simultaneous relationships, which depends on the degree of assortativity. Table 1 presents the fraction of persons that have simultaneous relationships cumulated over a 5-year period under varying choices for assortativity, ζ, and its corresponding κ. We find that under the random mixing model assumptions, 75%–85% of the population would have multiple relationships at some point during a 5-year period for values of concurrency κ in an interval from .17 to .26 that contained the estimate of .22 from Uganda [6]. The percentage is ≥85% under the disassortative mixing assumption for all values of ζ in the interval 0–0.99, although κ is higher, ranging from .66 to 2.00, in disassortative mixing compared to random mixing for the same pair formation and separation probabilities. It would be valuable to know the degree of disassortativity for Palombi et al and whether it leads to a reasonable concurrency level for the epidemic under consideration.

Table 1.

Cumulative Proportion of People Ever Having Concurrent Relationships

	Random Mixing		Disassortative Mixing
ζ	κ	Cumulative Proportion	κ	Cumulative Proportion
1	0	0	…	…
0.99	0.05	0.36	2	0.85
0.98	0.09	0.54	1.41	0.9
0.95	0.17	0.75	0.86	0.94
0.9	0.26	0.85	0.73	0.94
0.8	0.37	0.91	0.69	0.95
0.7	0.44	0.93	0.68	0.95
0.6	0.5	0.94	0.67	0.95
0.5	0.54	0.95	0.67	0.95
0	0.66	0.95	0.66	0.95

ζ is the tuning parameter for the level of mixing in the KM model, and κ is the corresponding index of concurrency.

The KM model also has implications for the total number of partners an individual has during a fixed period of time. Based on our simulations, the cumulative number of partners seems to follow a bell-shaped curve for both the random and disassortative mixing assumption, which is different from the highly skewed distributions that are typically associated with cumulative distributions of sexual partners [7].

In addition to the components in the simulation that control the dynamics of the evolving network, the probability of transmission per coital act stratified by plasma viral load can be very influential in the size and growth of the epidemic in the simulations. Palombi et al based the transmission probability on data obtained by Quinn et al [8], but 2 other groups [9, 10] have reported transmission probabilities that are substantially different. It would be useful to know the sensitivity of findings within a range of plausible transmission probabilities.

The KM epidemic model accommodates mixing by the number of current partners, but not by mixing patterns based on other covariates. Factors that affect mixing, including age, ethnicity, and socioeconomic status, have been shown to be influential in HIV transmission dynamics [4]. High levels of assortative mixing between groups can lead to multiple, almost independent epidemics [4]. Palombi et al model the impact of gradual implementation of a test-and-treat intervention at a district level and do not take into account the possibility that relationships can span across districts. High levels of relationships spanning districts that differ in the provision of the intervention will help determine its effectiveness.

In summary, the model of Palombi et al advances the work of Granich et al by incorporating many additional features, but readers would benefit from further clarification of underlying assumptions and characterization of their impact on results. Models intended to represent complex systems, such as those associated with the spread of HIV, inevitably have limitations, but understanding these limitations is essential if models are to be useful. Nonetheless, the consistency of the results obtained by Palombi et al and Granich et al provide additional evidence that identification and treatment of HIV-infected individuals should be a fundamental component of HIV control strategies.