Causal Inference Using Augmented Epidemic Models

TL;DR

This paper clarifies how time-varying interventions in epidemics can be analyzed causally by distinguishing augmented epidemic models as data-generating processes from their role as causal models. It demonstrates that standard maximum-likelihood or Bayesian estimation applied to augmented models yields the g-null paradox due to phantom variables, producing biased causal inferences. Four estimation strategies are analyzed, with Method 3 (marginal structural models via estimating equations using propensity scores) identified as the simplest and most natural approach, while Methods 2 and 4 offer principled alternatives that incorporate deeper causal structure. Through simulations and mobility-data applications, the authors show that MSM-based methods can provide consistent causal effects for time-varying interventions, enhance interpretability, and mitigate phantom-bias issues, albeit with modeling challenges in short time series and potential misspecification. The work has practical impact for scenario analysis and policy evaluation in epidemics, offering a principled framework for adjusting for confounders and time-varying interventions when only one or a few epidemic time series are available.

Abstract

Epidemic models describe the evolution of a communicable disease over time. These models are often modified to include the effects of interventions (control measures) such as vaccination, social distancing, school closings etc. Many such models were proposed during the COVID-19 epidemic. Inevitably these models are used to answer the question: What is the effect of the intervention on the epidemic? These models can either be interpreted as data generating models describing observed random variables or as causal models for counterfactual random variables. These two interpretations are often conflated in the literature. We discuss the difference between these two types of models, and then we discuss how to estimate the parameters of the model. Our focus is causal inference for parameters in epidemic models by adjusting for confounders, allowing time varying interventions.

Figures (8)

• Figure 1: Example DAG for epidemic data. The arrows indicate possible causal relationships between the outcome $Y$, intervention $A$ and confounders $X$. Latent variables $U$ are in pink; $U$ does not directly affect $A$ -- we say that $U$ is a phantom variable. If there were arrows from $U$ to $A$ then $U$ would instead be a confounder.
• Figure 2: Intervention graph from \ref{['fig::dagsemi0']} after setting $\overline{A}_t=\overline{a}_t$.
• Figure 3: Effect of phantoms. The latent phantom variable $U$ is not a confounder because it has no arrows to $A_0$ or $A_1$. Neither $A_0$ nor $A_1$ have a causal effect on $Y_1$. The variable $X_1$ is a collider, meaning that two arrowheads point to $X_1$. This implies that $Y_1$ and $(A_0, A_1)$ are dependent conditional on $X_1$, which in turn implies that the parameters that relate $Y_1$ to $(A_0, A_1)$ in the epidemic model will be non-zero even though there is no causal effect.
• Figure 4: Generating distribution $g$ from bhatt.
• Figure 5: Causal parameter estimation for semi-mechanistic epidemic model data. (a) Method 1: ML estimate and (b) Method 3: estimating equations estimate of $\beta_A$ averaged across 200 repeat simulations (blue dots) with 95%-confidence intervals (error bars), for a range of true $\beta_A$ values. There is phantom bias in (a) but not in (b). Small biases persist in (b), as evidenced by the confidence intervals excluding the true value of $\beta_A$ more often than they should. This is due to model misspecification, which we discuss in \ref{['sec::misspec']}.

Theorems & Definitions (10)

• Example 1: SIR Model
• Example 2: Discrete SEIR Model
• Example 3
• Example 4
• Example 5: Synthetic example
• Example 6: Semi-mechanistic Hawkes model
• Example 7: Semi-mechanistic Model
• Theorem 8
• Example 9: Semi-mechanistic Model
• Example 10: SEIR Model