{"abstract":"Transmission falls sharply after the first stage.","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":"Only the final stage is treated as infectious.","family":"w2-epidemic-erlang-stages-infectious-prevalence","id":"FA-65241","implementations":{"attempt":{"sha256":"66833ae265b7f5ed5b8d440ced72382e3241664599e68d6f7800ddce9d234287","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 = stages[-1]\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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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":"a6a40a8e49853d1531aeb411f48481b45c1aa117f9456d96783da48338fd3a36","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 = stages[0]\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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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":"36889ff3e3dd7c6c4b325a0ab02ed8cfe1f965793eab70279160bd5d7528c00b","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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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  ('control: no transmission', (0.0, 0.5, 2, 100, 10, 5), [[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-infectious-prevalence","generated_at":"2026-09-29T14:47:32.038433+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 infectious prevalence rule: `i = sum(stages)`.","root_cause":"Only stage 1 is treated as infectious.","sha256":"15f05949ad8a1548028a239a48ae73772fb4d2510e8dec89e5ed9848ee9509aa","title":"SIR with Erlang-distributed infectious period: infectious prevalence · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":39.108,"exit_code":1,"observations":[{"actual":[[10.0,9.967,9.797,9.518,9.189,8.849,8.514],8.358],"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":[[20.0,19.88,17.922,14.144,10.67],15.259],"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],9.662],"check":"control: no transmission","expected":[[10.0,7.361,3.917,1.837,0.805,0.338],9.662],"passed":true},{"actual":null,"check":"control: invalid 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