Marginal Likelihood Inference for Fitting Dynamical Survival Analysis Models to Epidemic Count Data

Technical Summary: Marginal Likelihood Inference for Fitting Dynamical Survival Analysis Models to Epidemic Count Data

Problem Statement
Stochastic compartmental models, such as the Stochastic SIR (Susceptible-Infected-Removed) model, are standard tools for describing disease spread. However, rigorous statistical inference under these models is often intractable when applied to real-world surveillance data. In practice, infection and recovery histories are rarely fully observed; instead, data typically arrive as discrete incidence counts (e.g., daily or weekly case reports). Under standard likelihood-based approaches, the marginal likelihood function for such partially observed data requires integrating over all possible unobserved trajectories consistent with the counts. This high-dimensional integration step is computationally prohibitive, often necessitating intensive simulation methods like particle filters, approximate Bayesian computation, or bespoke data-augmented Markov Chain Monte Carlo (MCMC) samplers.

Methodology
The authors propose a closed-form marginal likelihood for discretely observed incidence count data based on the Dynamical Survival Analysis (DSA) paradigm. The core of the method involves approximating the stochastic population-level hazard rate by its large-population limit while retaining a count-valued stochastic model.

1. DSA Framework: Building on Sellke's construction of the SIR model, the authors view the epidemic as an individual-level process where infection occurs when cumulative exposure exceeds a random threshold. In the large population limit ($N \to \infty$), the scaled susceptible and infected proportions converge to the solution of a system of Ordinary Differential Equations (ODEs).
2. Survival Interpretation: The authors interpret the solution to the ODEs as a survival function for the time to infection. This allows the derivation of explicit probability density functions for infection times.
3. Marginal Count Likelihood: The primary innovation is the derivation of a closed-form likelihood that exactly marginalizes over unknown infection and removal times.
   • For exact infection times (but missing recovery data), the likelihood is a product of densities.
   • For discrete count data (the primary focus), the authors derive a marginal likelihood by integrating out the unobserved infection times within discrete observation intervals. They demonstrate that under the DSA approximation, the joint density of the unobserved times and the observed counts factorizes such that the unobserved times follow a uniform distribution on the interval. Consequently, integrating them out yields a simple product form:
     $$l(\theta | \{Y_j\}_{j=1}^P) = s_T^{N-K} \prod_{j=1}^P (s_{\xi_{j-1}} - s_{\xi_j})^{Y_j}$$
     where $s_t$ is the solution to the ODE system at time $t$, $Y_j$ is the count in interval $j$, and $K$ is the total number of infections.
4. Generalizability: The framework is shown to extend to models with individual heterogeneity (frailty models) and heterogeneous mixing (Poisson random networks) by simply substituting the appropriate ODE solutions into the likelihood formula.

Key Contributions

• Closed-Form Likelihood: The paper provides the first closed-form likelihood for count data under the DSA approximation, eliminating the need for numerical integration over missing event times.
• Computational Efficiency: The method decouples computational cost from the population size $N$. Unlike exact methods where the grid resolution must increase with $N$, the DSA approach depends only on the number of observation times, making it scalable to large outbreaks.
• Flexibility: The inferential strategy is readily extensible to complex model variants, including those with frailty (unobserved heterogeneity in susceptibility) and network-based mixing, without requiring bespoke MCMC samplers.
• Connection to Interval Censoring: The derived likelihood is shown to be mathematically equivalent to an interval-censored survival analysis likelihood, providing a rigorous statistical foundation for the approximation.

Results
The authors evaluate the method through simulation studies and real-world case studies:

• Simulation Studies:

  • Accuracy: Parameter estimates using the marginal count likelihood are competitive with exact, computationally expensive methods (such as data-augmented MCMC) and significantly outperform methods that impute missing times as uniform random variables (which exhibit bias).
  • Coverage: While the DSA approximation leads to slightly reduced variability in trajectories compared to the exact stochastic process (resulting in credible interval coverage slightly below the nominal 95% level in some settings), the method achieves nominal coverage when data is simulated directly from the DSA approximation.
  • Efficiency: The DSA-based inference yields orders-of-magnitude improvements in computational speed (measured in effective samples per second) compared to exact methods, particularly as population size increases.
  • Frailty Models: The method successfully estimates parameters in frailty models, though practical identifiability issues arise with small populations and diffuse priors, which can be mitigated with more informative priors.

• Case Studies:

  • Ebola (DRC): Applied to the 2018–2020 Ebola outbreak using a Poisson random network model. The method, using only daily case counts, produced posterior means consistent with previous analyses that assumed knowledge of exact event times, but with wider credible intervals that better reflect the uncertainty of missing data.
  • COVID-19 (China): Applied to incidence data from 53 cities. The study compared a standard SIR model against a frailty model. The frailty model provided a superior fit to the data in several cities, suggesting that ignoring individual heterogeneity in susceptibility can lead to biased estimates of key epidemic parameters (e.g., $R_0$), even when population-level trends appear similar.

Significance and Claims
The authors claim that this work fills a critical methodological gap by enabling efficient, likelihood-based inference for stochastic epidemic models under realistic, partially observed data conditions. They emphasize that the approach offers a "user-friendly" alternative to simulation-heavy methods, allowing for straightforward posterior sampling using standard algorithms (e.g., Hamiltonian Monte Carlo) without bespoke model-based simulation.

The paper asserts that the DSA approximation provides a faithful representation of epidemic dynamics for large populations while offering a tractable likelihood structure. By demonstrating that the method extends to heterogeneity and network structures, the authors argue for a flexible modeling paradigm that balances computational feasibility with the ability to capture complex epidemic features. The work suggests that while the standard DSA-SIR model captures general trends, incorporating frailty or network structures is often necessary to explain residual variability in real-world data.