Technical Summary: DELENDA – Differentiable Epidemiology for Latent-state Estimation and Nonlinear Decision Analysis

Problem Statement

Subnational malaria tailoring (SNT) is fundamentally a population-level allocation problem: determining which interventions (e.g., insecticide-treated nets [ITNs], seasonal malaria chemoprevention [SMC], indoor residual spraying [IRS], and vaccines) should be prioritized, at what coverage levels, and under specific budget and uncertainty constraints.

Current workflows face a dichotomy in modeling approaches:

1. Simple models (e.g., Ross–Macdonald): Analytically tractable but lack the biological complexity (acquired immunity, age-specific risk, heterogeneity) necessary for nuanced policy.
2. Complex individual-based models (e.g., EMOD, OpenMalaria): Rich in biological detail but computationally expensive and often non-differentiable. This limits their utility in uncertainty-aware optimization, forcing analysts to rely on scenario sweeps or local sensitivity analysis rather than full posterior propagation.

The core bottleneck is that models capable of rigorous Bayesian calibration and continuous constrained optimization are often too coarse, while models with sophisticated biological representations are difficult to fit and integrate into decision loops that must account for uncertainty in vector biology, intervention efficacy, and costs.

Methodology

The DELENDA Model

DELENDA is a deterministic, structured compartmental model designed to bridge this gap. It tracks population mass across four key biological axes without simulating individual histories:

• Age: 11 bins spanning 0–75 years.
• Biting-risk heterogeneity: 5 strata derived from an exponential distribution.
• Immunity: Discretized into 10 levels.
• Multiplicity of Infection (MOI): Up to 15 concurrent infections.

The human state dimension is $5 \times 11 \times 10 \times 16 = 8,800$ compartments. The model couples this human subsystem with a seasonal vector subsystem (larvae, susceptible, exposed, and infectious mosquitoes).

Differentiable Implementation

The model is implemented entirely in JAX, making the forward simulation differentiable end-to-end.

• Solver: Uses an exponential-Euler update with mass-conservation corrections for stability.
• Gradients: Employs a custom vector–Jacobian product (VJP) first-order adjoint method for ODE integration, allowing reverse-mode differentiation through the solver.
• Workflow: The same compiled function is used for:
  1. Posterior Calibration: Using gradients for likelihood-based fitting.
  2. Decision Optimization: Using gradients for continuous constrained optimization over intervention coverage.

Calibration and Inference

• Data: Jointly calibrated to eight sites: five cross-sectional sites (Tanzania, Senegal, Kenya) and three pre-intervention villages from the Garki Project (Nigeria). Data includes age-stratified prevalence and clinical incidence.
• Likelihood: Uses beta-binomial for prevalence and negative-binomial for incidence.
• Inference: Uses Stochastic Variational Inference (SVI) for initialization and a No-U-Turn Sampler (NUTS) for the final posterior. The posterior consists of 38 parameters (30 shared natural-history/observation parameters and 8 site-specific EIR adjustments).

Decision Analysis Framework

The framework optimizes continuous coverage ($\theta_{int} \in [0, 0.9]^4$) for ITNs, SMC, IRS, and pediatric vaccination.

• Uncertainty Propagation: The analysis propagates uncertainty from four sources:
  1. The calibrated posterior (biological uncertainty).
  2. Vector-biology priors (conditional on observed EIR).
  3. Intervention-efficacy priors.
  4. Cost priors (resident-equivalent costs with convex overhead).
• Conditioning: To isolate vector uncertainty, the model conditions on a target Entomological Inoculation Rate (EIR) by solving for the larval recruitment amplitude ($K_{wet}$) for each posterior/vector draw.
• Optimization: Solves $\max_{\theta_{int}} E[\Delta Y]$ subject to budget constraints ($\rho[C] \le B$), where $\Delta Y$ is cases averted. Three budget rules are tested: Expected Cost, 90th Percentile Cost ($q_{90}$), and Conditional Value at Risk ($CVaR_{90}$).

Key Results

1. Calibration Performance

The NUTS posterior successfully reproduces age-stratified prevalence and clinical incidence across a broad EIR range (13 to 329) and matches Garki Project infectiousness targets. The posterior narrows central scale parameters while leaving some shape and maturation parameters weakly informed, providing a realistic uncertainty distribution.

2. Intervention Rankings and Robustness

In a highly seasonal transmission setting (modeled after Dapelogo, Burkina Faso), the optimized intervention rankings are robust to uncertainty:

• ITNs are prioritized first, reaching 90% coverage at budgets of \3–\5.
• SMC saturates rapidly for under-five objectives.
• Vaccines enter at moderate budgets.
• IRS appears later as a high-cost add-on.
• Uncertainty Impact: Adding posterior, vector, efficacy, and cost uncertainty changes the optimized coverage mix only modestly (mean L1 shift < 0.05). However, it substantially widens the distribution of projected cases averted (standard deviation of 6.5–7.3 percentage points).

3. Objective Sensitivity

The choice of objective function drives larger qualitative changes than uncertainty:

• Under-five optimization: Prioritizes SMC and vaccination earlier.
• All-age optimization: Delays vaccination and favors IRS, which protects the broader population via vector mortality.

4. Cost Uncertainty and Budget Rules

Cost uncertainty acts primarily as a constraint-side problem:

• Expected-cost optimization leads to a high probability (42–52%) of budget overruns.
• Tail-risk rules ($q_{90}$ and $CVaR_{90}$): These rules significantly reduce overrun probability (to ~10% and ~4.5%, respectively) but require sacrificing expected impact (8–11% reduction in cases averted). This trade-off allows programs to quantify the "operating margin" required for budget safety.

Significance and Claims

The paper positions DELENDA not as a replacement for individual-based simulators in all contexts, but as a specialized tool for population-level allocation problems where the decision target is an aggregate burden.

• Differentiability as a Workflow Enabler: The primary contribution is architectural. By making the compartmental model differentiable, DELENDA enables efficient Bayesian inference and continuous constrained optimization. This allows the full biological parameter space to be exposed to posterior calibration and carried into decision optimization, a task difficult with non-differentiable models.
• Uncertainty-First Approach: The framework demonstrates that for many subnational tailoring questions, the ranking of interventions is more robust than the projected impact. The value of the approach lies in quantifying the width of the impact distribution around a stable optimal portfolio, rather than constantly shifting the portfolio itself.
• Operational Relevance: The paper claims to provide a reusable workflow for subnational tailoring that explicitly couples the estimation problem (calibration) with the decision problem (optimization), ensuring that recommendations are robust to the assumptions that generated them.

The authors explicitly state that the results are a "proof-of-concept" and a "methodological case study" rather than a deployment-ready recommendation for a specific district, noting that operational use would require local inputs for costs, resistance, and implementation constraints.