Graphical Models of Pandemic

Abstract

Both COVID-19 and novel pandemics challenge those of us within the modeling community, specifically in establishing suitable relations between lifecycles, scales, and existing methods. Herein we demonstrate transitions between models in space/time, individual-to-community, county-to-city, along with models for the trace beginning with exposure, then to symptom manifest, then to community transmission. To that end, we leverage publicly available data to compose a chain of Graphical Models (GMs) for predicting infection rates across communities, space, and time. We'll anchor our GMs against the more expensive yet state-of-the-art Agent-Based Models (ABMs). Insight obtained from designing novel GMs calibrated to ABMs furnishes reduced, yet reliable surrogates for the end-to-end public health challenge of community contact tracing and transmission. Further, this novel research transcends and synergizes information integration and informatics, leading to an advance in the science of GMs. Cognizance into the data lifecycle using properly coarse-grained modeling will broaden the toolkit available to public health specialists, and hopefully empower governments and health agencies, here and abroad, in addressing the profound challenges in disease and vaccination campaigns confronting us by COVID and future pandemics. In this proof of principle study, focusing on the GM methodology development, we show, first, how static GM of the Ising model type (characterised by pair-wise interaction between nodes related to traffic and communications between nodes representing communities, or census tracts within a given city, and with local infection bias) emerge from a dynamic GM of the Independent Cascade type, introduced and studied in Computer and Networks sciences mainly in the context of the spread of social influences. Second, we formulate the problem of inference in epidemiology as inference problems in the Ising model setting. Specifically, we pose the challenge of computing Conditional A-posteriori Level of Infection (CALI), which provides a quantitative answer to the questions: What is the probability that a given node in the GM (given census tract within the city) becomes infected in the result of injection of the infection at another node, e.g. due to arrival of a super-spreader agent or occurrence of the super-spreader event in the area. To answer the question exactly is not feasible for any realistic size (larger than 30-50 nodes) model. We therefore adopt and develop approximate inference techniques, of the variational and variable elimination types, developed in the GM literature. To demonstrate utility of the methodology, which seems new for the public health application, we build a 123-node model of Seattle, as well as its 10-node and 20-node coarse-grained variants, and then conduct the proof of principles experimental studies. The experiments on the coarse-grained models have helped us to validate the approximate inference by juxtaposing it to the exact inference. The experiments also lead to discovery of interesting and most probably universal phenomena. In particular, we observe (a) a strong sensitivity of CALI to the location of the initial infection, and (b) strong alignment of the resulting infection probability (values of CALI) observed at different nodes in the regimes of moderate interaction between the nodes. We then speculate how these, and other observations drawn from the synthetic experiments, can be extended to a more realistic, data driven setting of actual operation importance. We conclude the manuscript with an extensive discussion of how the methodology should be developed further, both at the level of devising realistic GMs from observational data (and also enhancing it with microscopic ABM modeling and simulations) and also regarding utilisation of the GM inference methodology for more complex problems of the pandemic mitigation and control.

Prediction and Prevention of Pandemics via Graphical Model Inference and ConvexProgramming

“…First, we consider the case when the only node a = 1 is infected. In this simple setting there are four possible MAP states, (x 1 , x 2 , x 3 ) ∈ {(+1, −1, −1), (+1, −1, +1), (+1, +1, −1), (+1, +1, +1)}, shown in (1). We observe that the two "polarized" MAP states, (+1, −1, −1) and (+1, +1, +1), are seen most often among the samples, while domain occupied by the other two "mixed" MAP states, (+1, −1, +1) and (+1, +1, −1) is much smaller, with the two modes positioned on the interface between the two polarized states.…”

“…of the Independent Cascade Model (ICM) type [24][25][26][27][28] , were introduced even earlier in the CS/AI literature in the context of modeling how the rumors spread over social networks (with a side reference on using ICM in epidemiology). As argued in 1 the Independent Cascade Models (ICMs) can be adapted to modeling pandemics. (Another interesting use of the ICM to model COVID-19 pandemic was discussed in 29 .)…”

Prediction and prevention of pandemics via graphical model inference and convex programming

“…We follow our previous work 1 in justification for the use of the Graphical Models (GM) to study and mitigate pandemics. Therefore, we start from providing a brief recap of the prior literature on modeling of the epidemics, describe the logic which led us in 1 to the Ising Model (IM) formulation, and then state formally the inference and prevention problems addressed in the manuscript.…”

“…of the Independent Cascade Model (ICM) type 24 – 28 , were introduced even earlier in the CS/AI literature in the context of modeling how the rumors spread over social networks (with a side reference on using ICM in epidemiology). As argued in 1 the Independent Cascade Models (ICMs) can be adapted to modeling pandemics. (Another interesting use of the ICM to model COVID-19 pandemic was discussed in 29 .)…”

“…It was shown in 1 that with some regularization applied, statistics of the terminal state of the Cascade Model of Pandemic turns into a Graphical Model of the attractive Ising Model type.…”