Accounting for contact network uncertainty in epidemic inferences with Approximate Bayesian Computation

Abstract

In models of infectious disease dynamics, the incorporation of contact network information allows for the capture of the non-randomness and heterogeneity of realistic contact patterns. Oftentimes, it is assumed that this underlying network is known with perfect certainty. However, in realistic settings, the observed data usually serves as an imperfect proxy of the actual contact patterns in the population. Furthermore, event times in observed epidemics are not perfectly recorded; individual infection and recovery times are often missing. In order to conduct accurate inferences on parameters of contagion spread, it is crucial to incorporate these sources of uncertainty. In this paper, we propose the use of Network-augmented Mixture Density Network-compressed ABC (NA-MDN-ABC) to learn informative summary statistics for the available data. This method will allow for Bayesian inference on the parameters of a contagious process, while accounting for imperfect observations on the epidemic and the contact network. We will demonstrate the use of this method on simulated epidemics and networks, and extend this framework to analyze the spread of Tattoo Skin Disease (TSD) among bottlenose dolphins in Shark Bay, Australia.

Keywords: Approximate Bayesian Computation; Networks; SIR model.

Algorithm 1

Rejection ABC

Algorithm 2

Network-Augmented Rejection ABC

Fig. 1

Using the framework from Young et al. (2020), network samples $A^{\prime }$ are drawn based on observed network data X. Proposal contagion parameter values ${\theta }^{\prime }$ are drawn from the prior. $A^{\prime }$ and ${\theta }^{\prime }$ are passed through the epidemic model to generate a simulated output $Y^{\prime }$. A summary statistic of $Y^{\prime }$ is calculated via an MDN, yielding $S(Y^{\prime })$. Finally, $S(Y^{\prime })$ is compared to the summary statistics yielded by the original observed epidemic Y. If the acceptance condition is fulfilled, such that $d(S(Y^{\prime }),S(Y))\le \delta$, ${\theta }^{\prime }$ and $A^{\prime }$ are accepted as part of the MDN-ABC posterior. To generate additional posterior samples, simply repeat the process with new values of ${\theta }^{\prime }$ and $A^{\prime }$

Fig. 2

Posterior samples generated from NA-MDN-ABC. Panels b–e display the results for the Erdős–Rényi network, and panels f–j show results for the log-normal distributed network. “True” values are marked with a vertical line. Prior densities are shown in gray (prior densities for ${\lambda }_0$ and ${\lambda }_1$ have been multiplied by an additional factor of 10 for visibility)

Fig. 3

NA-MDN-ABC results across 10 instances of the original stochastic epidemic for $\beta$ and $\gamma$ on a Erdős–Rényi network (a and c) and for $\beta$ and $\gamma$ on a log-normal network (b and d). True values for parameters are marked with a horizontal line

Fig. 4

Empirical coverage of credible intervals plotted against nominal coverages of the MDN-ABC step for a) $\beta$ and c) $\gamma$ for the Erdős–Rényi network, and b) $\beta$ and d) $\gamma$ for the Log-normal network

Fig. 5

In Panel (a), discrepancy to the observed data is plotted against discrepancy to data generated from the model, with the dotted line denoting equality. In panels (b)–(e), samples from 10 independent HMC chains are shown for b $n_0$, c $n_1$, d $p_0$, and e $\rho$, for time $w=0$

Fig. 6

NA-MDN-ABC approximate posterior densities for contagion parameters, a ${\beta }_c$, b ${\beta }_j$, c ${\beta }_a$, d $ϵ$, e ${\gamma }_a$, f ${\gamma }_b$

Fig. 7

a Posterior distribution of contact network among dolphins, for time $w=2$. Thicker edge weights and shades correspond to higher edge probabilities. Only edges with posterior probability greater than $10\%$ were included. The color of each node corresponds to the age category of the dolphin: adults in blue, juveniles in red, and calves in green. b Observed number of interactions between dolphins, with thicker edge weights and shades corresponding to higher counts of co-proximity events

Fig. 8

a NA-MDN-ABC approximate posterior density for infectious period. b Posterior predictive distribution of the mean difference between first and last infectious sightings, compared to Powell et al. (2018). c Predictive intervals for cumulative distribution function of initial infection times. d Predictive intervals for cumulative distribution function of interval between first and last infected sightings, for dolphins who were spotted with symptoms at least twice

Fig. 9

Posterior samples generated from NA-MDN-ABC, utilizing SMC-ABC instead of rejection ABC. Panels b–e display the results for the Erdős–Rényi network, and panels f–j show results for the log-normal distributed network. “True” values are marked with a vertical line. Prior densities are shown in gray (prior densities for ${\lambda }_0$ and ${\lambda }_1$ have been multiplied by an additional factor of 10 for visibility)

Fig. 10

Trace plots for network parameters $\varphi$, across all 4 years of observation