{"abstract":"Recovered counts rise immediately at stage-1 exit rates.","category":"Epidemic compartment models","checks":7,"contract":"Infectious period split into k sequential stages each left at rate k*gamma; seeds start in stage 1; all stages transmit; Euler with dt=0.1 (10 substeps/day); return [total I at day ends including day 0 rounded 3, R rounded 3], None for k<1.","evaluation_group":"w2-epidemic-erlang-stages","failed_approach":"Averaging all stage flows is not the final-stage exit.","family":"w2-epidemic-erlang-stages-removal-stage","id":"FA-65231","implementations":{"attempt":{"sha256":"168c8ace5273cc08746992e589fe8cbec07f8eb7758bf7678e55df95781aecf0","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(beta, gamma, k, pop, i0, days):\n    if k < 1:\n        return None\n    stages = [0.0] * k\n    stages[0] = float(i0)\n    s = float(pop - i0)\n    r = 0.0\n    dt = 0.1\n    series = [round(sum(stages), 3)]\n    for _ in range(days):\n        for _ in range(10):\n            i = sum(stages)\n            inf = beta * s * i / pop * dt\n            flows = [k * gamma * x * dt for x in stages]\n            s -= inf\n            stages[0] += inf - flows[0]\n            for j in range(1, k):\n                stages[j] += flows[j - 1] - flows[j]\n            r += sum(flows) / k\n        series.append(round(sum(stages), 3))\n    return [series, round(r, 3)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"f4a7f025c4789ca707300df36e7984aa27df7e9969d81d278bb2cf2dc4e3dc4d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(beta, gamma, k, pop, i0, days):\n    if k < 1:\n        return None\n    stages = [0.0] * k\n    stages[0] = float(i0)\n    s = float(pop - i0)\n    r = 0.0\n    dt = 0.1\n    series = [round(sum(stages), 3)]\n    for _ in range(days):\n        for _ in range(10):\n            i = sum(stages)\n            inf = beta * s * i / pop * dt\n            flows = [k * gamma * x * dt for x in stages]\n            s -= inf\n            stages[0] += inf - flows[0]\n            for j in range(1, k):\n                stages[j] += flows[j - 1] - flows[j]\n            r += flows[0]\n        series.append(round(sum(stages), 3))\n    return [series, round(r, 3)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"147f167b1673c9402d4dd9a045605c63c60b05540ca136797ab231ba52f3d3c5","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(beta, gamma, k, pop, i0, days):\n    if k < 1:\n        return None\n    stages = [0.0] * k\n    stages[0] = float(i0)\n    s = float(pop - i0)\n    r = 0.0\n    dt = 0.1\n    series = [round(sum(stages), 3)]\n    for _ in range(days):\n        for _ in range(10):\n            i = sum(stages)\n            inf = beta * s * i / pop * dt\n            flows = [k * gamma * x * dt for x in stages]\n            s -= inf\n            stages[0] += inf - flows[0]\n            for j in range(1, k):\n                stages[j] += flows[j - 1] - flows[j]\n            r += flows[-1]\n        series.append(round(sum(stages), 3))\n    return [series, round(r, 3)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('regression: no transmission',\n   (0.0, 0.5, 2, 100, 10, 5),\n   [[10.0, 7.361, 3.917, 1.837, 0.805, 0.338], 9.662]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])],\n [('regression: three stages',\n   (0.5, 0.2, 3, 1000, 10, 6),\n   [[10.0, 15.964, 24.203, 35.419, 50.774, 71.593, 99.086], 25.589]),\n  ('control: single stage equals SIR',\n   (0.4, 0.25, 1, 500, 5, 5),\n   [[5.0, 5.776, 6.66, 7.664, 8.798, 10.074], 9.032]),\n  ('regression: many stages', (0.6, 0.3, 5, 800, 20, 4), [[20.0, 34.937, 55.691, 80.709, 110.985], 33.142]),\n  ('control: invalid k', (0.5, 0.2, 0, 100, 5, 3), None),\n  ('regression: two stages slow',\n   (0.3, 0.1, 2, 2000, 30, 7),\n   [[30.0, 39.545, 50.929, 64.696, 81.433, 101.776, 126.384, 155.912], 30.058]),\n  ('control: boundary zero days', (0.5, 0.2, 3, 100, 5, 0), [[5.0], 0.0]),\n  ('regression: four stages fast recovery',\n   (0.9, 0.5, 4, 300, 6, 5),\n   [[6.0, 12.89, 21.627, 33.085, 46.281, 57.492], 51.385])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"Deterministic bounded teaching model with a stipulated contract; not a validated scientific or public-health modelling library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-epidemic-erlang-stages-removal-stage","generated_at":"2026-09-29T14:47:31.994938+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Compartmental epidemic calculations drive outbreak forecasts, vaccine targets and hospital planning; a single wrong flow, rate conversion or boundary silently changes every downstream number.","repair":"Restore the removal stage rule: `r += flows[-1]`.","root_cause":"Removals are taken from the first stage instead of the last.","sha256":"0104863d075b76c3caa163f5799d5487e12a84161fa047849bd51c781c5f2d97","title":"SIR with Erlang-distributed infectious period: removal stage · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":39.699,"exit_code":1,"observations":[{"actual":[[10.0,15.964,24.203,35.419,50.774,71.593,99.086],49.175],"check":"regression: three stages","expected":[[10.0,15.964,24.203,35.419,50.774,71.593,99.086],25.589],"passed":false},{"actual":[[5.0,5.776,6.66,7.664,8.798,10.074],9.032],"check":"control: single stage equals SIR","expected":[[5.0,5.776,6.66,7.664,8.798,10.074],9.032],"passed":true},{"actual":[[20.0,34.937,55.691,80.709,110.985],69.149],"check":"regression: many stages","expected":[[20.0,34.937,55.691,80.709,110.985],33.142],"passed":false},{"actual":[[10.0,7.361,3.917,1.837,0.805,0.338],9.805],"check":"regression: no transmission","expected":[[10.0,7.361,3.917,1.837,0.805,0.338],9.662],"passed":false},{"actual":null,"check":"control: invalid 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