Approximate inference for longitudinal mechanistic HIV contact network

Abstract

Network models are increasingly used to study infectious disease spread. Exponential Random Graph models have a history in this area, with scalable inference methods now available. An alternative approach uses mechanistic network models. Mechanistic network models directly capture individual behaviors, making them suitable for studying sexually transmitted diseases. Combining mechanistic models with Approximate Bayesian Computation allows flexible modeling using domain-specific interaction rules among agents, avoiding network model oversimplifications. These models are ideal for longitudinal settings as they explicitly incorporate network evolution over time. We implemented a discrete-time version of a previously published continuous-time model of evolving contact networks for men who have sex with men and proposed an ABC-based approximate inference scheme for it. As expected, we found that a two-wave longitudinal study design improves the accuracy of inference compared to a cross-sectional design. However, the gains in precision in collecting data twice, up to 18%, depend on the spacing of the two waves and are sensitive to the choice of summary statistics. In addition to methodological developments, our results inform the design of future longitudinal network studies in sexually transmitted diseases, specifically in terms of what data to collect from participants and when to do so.

Keywords: ABC; Agent based modeling; HIV; Inference; MSM; Mechanistic model; Networks.

Fig. 1

Network visualizations containing cumulative (from iteration (1) steady (red dashed) and casual (blue solid) edges for iterations 1 (left), 6 (middle), and 12 (right). We used the following parameter values: $\mu$ = 0, $\rho$ = 0.3, $\sigma$ = 0.1, $w_1$ = 0.2, $w_0$ = 0.4

Algorithm 1

Hansson MSM model (Hansson et al. 2019)

Fig. 2

Prior distributions on the inverse parameters and the corresponding implied prior distributions on parameters themselves. The top row shows the distributions of the inverse parameters, which can be interpreted as distributions of the average values of geometric distributions. The bottom row shows the distributions of the parameter values themselves for our discrete time mechanistic network model for the following parameters: $\rho$, $\sigma$, ${\omega }_1$, ${\omega }_0$

Fig. 3

Pairwise relationships between the model parameters (horizontal axes) and the summary statistics (vertical axes) used in our ABC inference scheme. Free parameters are fixed at $\mu$ = 0, $\rho$ = 0.3, $\sigma$ = 0.1, ${\omega }_0$ = 0.4, ${\omega }_1$ = 0.2. The lag between two consecutive network observations is fixed at 15 iterations. Each box plot consists of 100 samples

Fig. 4

Pairwise relationships between the model parameters (horizontal axes) and the summary statistics (vertical axes) used in our ABC inference scheme. Free parameters are sampled from the prior distributions. The lag between two consecutive network observations is fixed at 15 iterations. Each box plot consists of 100 samples

Fig. 5

Approximate marginal posterior distributions of model parameters obtained by retaining the top 1% of proposed prior samples in our ABC accept/reject inference scheme. Different rows correspond to comparing the prior (top), observing the graph once (middle), and observing the graph twice with a lag of 50 iterations (bottom). All posteriors include a regression adjustment. The blue solid lines represent the 95% credible intervals and the red dotted lines represent the true parameter values

Fig. 6

Estimated average RMSE, where the average is taken across multiple network realizations, as a function of the lag between the two network observations. We also include a loess curve with a 95% confidence interval (shaded areas). The average prior average RMSE is 2.22 (not shown), whereas the corresponding regression adjusted error for a network observed only once is 1.03 that for a network observed twice with a lag of 50 iterations is 0.84

Fig. 7

Estimated regression adjusted average RMSE for the total error (top curve) and separately for the four parameters considered in our study (bottom four curves). These results show that when observing a network twice, the reduction in total RMSE is mainly due to the reduction of RMSE for $\rho$ and $\sigma$