Stan ODE Modeler

Use this skill when writing Stan models where latent dynamics are defined by ODEs: epidemic models (SIR/SEIR), PK/PD compartment models, population dynamics, biochemical reactions, growth models.

Uses Stan 2.24+ modern ODE interface (ode_rk45, ode_bdf, ode_adams, adjoint solvers), not legacy integrate_ode_*.

Workflow

1. Extract generative model: Identify state variables x(t), parameters θ, ODE system dx/dt = f(t,x,θ), observation model, data structure
2. Choose solver (see below)
3. Write complete Stan program: functions (ODE RHS), data, parameters, transformed parameters (solve ODE), model (priors + likelihood), generated quantities (predictions)
4. Scale appropriately: Non-dimensionalize time and states to O(1) scales, use log/logit transforms for constraints
5. Use stan-coding skill for general Stan structure, parameterization, and ArviZ integration

Modern ODE Interface

Function signature:

functions {
  vector ode_rhs(real t, vector y, array[] real theta, array[] real x_r, array[] int x_i) {
    vector[N] dydt;
    // fill dydt based on y and theta
    return dydt;
  }
}

Solver calls:

• ode_rk45(ode_rhs, y0, t0, ts, theta, x_r, x_i) - non-stiff (default)
• ode_bdf(ode_rhs, y0, t0, ts, theta, x_r, x_i) - stiff systems
• ode_adams(ode_rhs, y0, t0, ts, theta, x_r, x_i) - smooth, long horizons
• ode_adjoint_* - many parameters relative to state dimension

Add _tol suffix for configurable tolerances: ode_rk45_tol(..., rel_tol, abs_tol, max_num_steps, ...)

Returns: array[T] vector[N] where T = length(ts), N = state dimension

Program Structure

Transformed parameters (when ODE enters likelihood):

transformed parameters {
  array[T] vector[N] x_hat;
  array[] real theta = {beta, gamma};  // pack parameters
  x_hat = ode_rk45(ode_rhs, y0, t0, ts, theta, {}, {});
}

Generated quantities (predictions, forecasts): Include y_rep and log_lik following stan-coding skill guidelines.

Example: SIR Model

functions {
  vector sir_rhs(real t, vector y, array[] real theta, array[] real x_r, array[] int x_i) {
    vector[3] dydt;
    real beta = theta[1];
    real gamma = theta[2];
    real N_pop = x_r[1];

    dydt[1] = -beta * y[1] * y[2] / N_pop;  // dS/dt
    dydt[2] = beta * y[1] * y[2] / N_pop - gamma * y[2];  // dI/dt
    dydt[3] = gamma * y[2];  // dR/dt
    return dydt;
  }
}

transformed parameters {
  array[T] vector[3] x_hat;
  x_hat = ode_rk45(sir_rhs, y0, t0, ts, {beta, gamma}, {N_pop}, {});
}

model {
  beta ~ lognormal(0, 1);
  gamma ~ lognormal(0, 1);
  for (t in 1:T) {
    I_obs[t] ~ poisson(x_hat[t, 2]);  // observe infected count
  }
}

Hierarchical ODE Models

For multiple subjects with subject-specific ODE parameters:

• Use population hyperparameters (mean, sd) in parameters block
• Draw subject-level parameters with non-centered parameterization
• Loop over subjects, solve ODE once per subject
• Use log-transforms for positive parameters: r[n] = exp(mu_log_r + sigma_log_r * z_r[n])

Solver Selection

Default: ode_rk45 with rel_tol ≈ 1e-6, abs_tol ≈ (typical state scale) × 1e-6, max_num_steps ≈ 1e4-1e5

Switch to ode_bdf if:

• "max_num_steps exceeded" warnings
• Fast/slow components (multi-compartment PK, stiff reactions)
• Rapid changes in some states while others change slowly

Use ode_adams if: Smooth, non-stiff systems over long time horizons

Use ode_adjoint_* if: Many parameters relative to state dimension (hierarchical models with per-subject parameters)

Tolerances: ODE error should be much smaller than observation noise. Rarely need rel_tol < 1e-8. Can relax slightly (1e-6 → 1e-5) if HMC is slow but stable.

Debugging Common Issues

Max steps exceeded:

• Rescale time and states to O(1)
• Switch to ode_bdf if stiff
• Increase max_num_steps as last resort

Divergences:

• Use log-transforms for positive parameters
• Use convergence-diagnostics skill for HMC-specific issues
• Strengthen priors to avoid extreme parameter values

Performance:

• Use observation times only, not fine grids
• Consider reduce_sum parallelization for multi-subject models
• Ensure proper scaling (time and states near 1)

References

For ODE model examples and patterns, search:

• Stan case studies: https://mc-stan.org/learn-stan/case-studies.html (search for PK/PD, epidemiology, ODE examples)
• Stan ODE documentation: https://mc-stan.org/docs/functions-reference/functions-ode-solver.html

Use WebSearch or WebFetch to find domain-specific ODE modeling examples.