## Problem 057 — rank:24

**Title:** Host–microbiome system identification

**Added binding assumptions (LaTeX):**

```latex
With fixed parameters, initial state and intervention, the displayed ODE model is deterministic and $p_\theta$ is only zero or one. For a nontrivial probability supply a random initial-state distribution, process-noise law or explicitly modeled subject heterogeneity. Observation noise alone does not randomize the true adverse-state event. Restrict the polynomial dynamics to a forward-complete positive domain or report model blow-up as failure.
```

**Direct edits:** 0. **Fresh external checks:** None; supplied sources retained as provenance.