# ⚛  L1 Principle — SIR Epidemic Parameter Estimation

**ID:** `L1-405` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Epidemiology*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** SIR parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** negative binomial
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555218

---

## 🧠 1. Introduction

**SIR Epidemic Parameter Estimation** is a **parameter-estimation problem** whose unknown lives in **SIR parameter vector** space, within the **Epidemiology** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **incidence case surveillance** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · likelihood · incidence nb operator; S · mcmc · epidemic posterior operator.

Observations are corrupted by negative-binomial over-dispersed counts. Existence of the recovered SIR_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); under_reporting_factor dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Negative binomial sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Time derivative → O · likelihood · incidence nb → S · mcmc · epidemic posterior → **y** (detector).

```
y = `S.mcmc.epidemic_posterior` `O.likelihood.incidence_nb` ∂_t x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.likelihood.incidence_nb` | O · likelihood · incidence nb operator |
| `S.mcmc.epidemic_posterior` | S · mcmc · epidemic posterior operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Epidemiology |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | SIR_parameter_vector |
| Noise model | negative_binomial |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered SIR_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); under_reporting_factor dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Negative binomial sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | R0_estimation_RMSE |
| Secondary | incidence_prediction_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `incidence_case_surveillance`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| R0 true | — | 2.5 |
| Detection rate | — | 0.5 |
| N incidence days | — | 100 |
| Overdispersion k | — | 10 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| R0 true | — | 1.0 – 10.0 |
| Detection rate | — | 0.05 – 1.0 |
| N incidence days | — | 10 – 500 |
| Overdispersion k | — | 1 – 100 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.2 R0_estimation_RMSE

| Metric | Range |
|---|---|
| R0 estimation rmse | 0.05 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **R0_estimation_RMSE**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x4d92422c8c3788c9c3d985065862fc4d9515767c7738699528c989f329842e4c`
- **Chain tx hash:** `0x395004fb80e2a73bc39abc039c5bd894b87130fdc19cbfbedda6f2aa2d99083b`
- **Chain block:** `41555218`

---

## File Mapping

This bundle consists of: `L1-405.md`, `L1-405.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-405.md` | Source of truth — edit this | Human or LLM |
| `L1-405.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-405`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._