A Complete-Data Likelihood for Epidemic Processes on Partially Observed Dynamic Networks

Technical Summary: A Complete-Data Likelihood for Epidemic Processes on Partially Observed Dynamic Networks

Problem Statement
Inference for infectious disease transmission on dynamic contact networks is fundamentally complicated by four interrelated challenges: latent infection times, partially observed network evolution, measurement error in contact data, and infection originating from outside the observed population (external infection pressure). Existing likelihood-based approaches typically address these issues in isolation, often relying on restrictive assumptions such as fully observed networks, closed populations, or the use of symptom onset as a direct surrogate for infection time. These simplifications obscure important questions regarding parameter identifiability and uncertainty propagation, particularly when both disease trajectories and network evolution are only partially observed. The observed data represent a coarse projection of an underlying stochastic process, requiring simultaneous treatment of latent disease states, latent contact histories, and imperfect observation mechanisms.

Methodology
The paper proposes a unified complete-data likelihood framework that models disease progression, network evolution, and observation mechanisms as interacting continuous-time stochastic processes within a single probabilistic structure.

1. Joint Stochastic Model: The framework couples a Susceptible–Exposed–Infectious–Removed (SEIR) epidemic process with a status-dependent dynamic contact network.

   • Epidemic Process: Modeled as a continuous-time Markov chain where transition hazards depend on the current network configuration. Specifically, the hazard for a susceptible individual $i$ to become exposed is $\lambda_{SE}^i(t) = \beta I_i(t) + \xi$, where $\beta$ is the within-network transmission rate, $I_i(t)$ is the number of infectious contacts, and $\xi$ is the external infection rate. Progression from Exposed to Infectious ($\kappa$) and Infectious to Removed ($\gamma$) follows constant rates.
   • Dynamic Network: The contact network evolves as a continuous-time Markov process where edge formation ($\eta_{ab}$) and dissolution ($\tau_{ab}$) rates depend on the disease states ($a, b$) of the incident nodes. This allows for feedback loops where infection status alters contact behavior.
   • Observation Model: Explicit models are introduced for symptoms and contacts. Symptom observations are subject to misclassification (sensitivity/specificity), and contact observations are modeled with error (false positives/negatives), acknowledging that data are often discrete, noisy, and intermittent.

2. Complete-Data Likelihood Derivation:
   The core contribution is the derivation of a complete-data event-history likelihood for the joint epidemic–network process. By treating the full latent trajectory (infection times, progression times, and edge histories) as missing data, the authors derive a likelihood function that factorizes into:

   • A latent process component (product of transition intensities and exponential of integrated hazards).
   • An observation component (product of conditional probabilities of observed data given latent states).

   This formulation yields closed-form Maximum Likelihood Estimators (MLEs) for all parameters (epidemic rates, network rates, and observation parameters) conditional on the latent path. These estimators take the familiar "event-count-over-exposure" form.

3. Inferential Framework:

   • Frequentist: The paper establishes the asymptotic normality of the complete-data MLEs and derives closed-form expressions for the observed and expected Fisher information matrices, facilitating standard error calculations.
   • Bayesian: Recognizing that the observed-data likelihood is intractable due to the combinatorial explosion of latent paths, the paper proposes a Bayesian data augmentation strategy. A Metropolis-within-Gibbs sampler is constructed to alternately update the latent epidemic history, the latent network trajectory, and the model parameters. This approach leverages the conditional conjugacy of the complete-data likelihood (e.g., Gamma priors for rate parameters, Beta priors for observation probabilities) to facilitate efficient posterior sampling.

Key Contributions

• Unified Framework: The paper provides a general likelihood framework that separates the latent process from the observation process while preserving their statistical dependence, accommodating latent incubation periods, intermittent network observation, and external infection pressure simultaneously.
• Derivation of Complete-Data Likelihood: It derives a rigorous complete-data event-history likelihood that serves as a foundation for both likelihood-based and Bayesian inference via data augmentation.
• Identifiability Analysis: The work clarifies how information from disease progression and contact dynamics jointly determines parameter estimability. It distinguishes between structural identifiability (under complete observation) and practical identifiability (under partial observation), revealing that parameters governing internal vs. external infection and dynamic contact formation are particularly vulnerable to weak identifiability when data are sparse.
• Special Cases: The framework is shown to encompass several existing epidemic network models as special cases (e.g., static networks, closed populations, fully observed networks).

Results
A simulation study was conducted with $N=100$ individuals across three observation regimes: High, Moderate, and Sparse.

• Parameter Recovery: Under High and Moderate observation regimes, the proposed Bayesian sampler accurately recovered epidemic rates ($\beta, \kappa, \gamma$), network rates ($\eta, \tau$), and observation parameters ($p, s, c$) with low bias and well-calibrated 95% credible intervals.
• Sparse Observation: Under sparse observation, the method remained stable. While the posterior uncertainty for network parameters and the external infection rate ($\xi$) increased, the estimates remained centered near the truth. The credible intervals widened appropriately to reflect the loss of information, avoiding overconfident inference.
• Confounding: The study confirmed that internal transmission ($\beta$) and external infection ($\xi$) are difficult to separate without rich contact data. Similarly, network formation parameters were highly sensitive to missingness and measurement error.
• Latent Reconstruction: The model successfully propagated uncertainty, reconstructing plausible latent epidemic and network trajectories consistent with the observed data, even when direct observation was limited.

Significance and Claims
The paper claims to establish a foundation for uncertainty-aware analysis of complex transmission processes on evolving networks. Its primary significance lies in providing a coherent probabilistic structure for inference in settings where data are inherently incomplete and noisy—a common reality in infectious disease epidemiology.

The authors assert that their framework:

1. Clarifies Estimability: It reveals how the joint observation of symptoms and contacts is necessary to disentangle transmission dynamics from network dynamics and external infection pressures.
2. Enables Joint Inference: It allows for the simultaneous learning of model parameters and latent histories, avoiding the biases introduced by treating network or infection times as fixed or fully observed.
3. Provides Robustness: The simulation results demonstrate that the method yields accurate point estimates and sensible posterior uncertainty even when data quality is suboptimal, making it a viable tool for real-world applications where perfect data is unavailable.

The paper concludes that while the method performs best when both symptom and contact data are moderately informative, it offers a flexible and effective tool for inference in partially observed epidemic systems, bridging the gap between mechanistic modeling and statistical inference.