FAILURE MAP
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FA-64851 / Epidemic compartment models / Open access

SIR Euler integrator with daily sampling: recovery step size · case 01

Recovery runs substeps times too fast whenever the day is subdivided.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The recovery flow omits the dt factor while infection applies it.

VERIFIED REPAIR

Restore the recovery step size rule: `new_rec = gamma * i * dt`.

Unsuccessful approach: Using the seed prevalence i0 makes recovery independent of current prevalence.

Case contract

Frequency-dependent SIR integrated with forward Euler at dt=1/substeps; new infections per substep are beta*S*I/pop*dt capped at S; peak prevalence and its day are sampled at day ends (day 0 is the seed); return [S, peak I, peak day, R] rounded to 3 decimals, or None for invalid input.

Why this case matters

Compartmental epidemic calculations drive outbreak forecasts, vaccine targets and hospital planning; a single wrong flow, rate conversion or boundary silently changes every downstream number.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(beta, gamma, pop, i0, days, substeps):
    if pop <= 0 or i0 < 0 or i0 > pop or substeps < 1:
        return None
    dt = 1.0 / substeps
    s = float(pop - i0)
    i = float(i0)
    r = 0.0
    peak_i = i
    peak_day = 0
    for day in range(1, days + 1):
        for _ in range(substeps):
            new_inf = min(beta * s * i / pop * dt, s)
            new_rec = gamma * i
            s -= new_inf
            i += new_inf - new_rec
            r += new_rec
        if i > peak_i:
            peak_i = i
            peak_day = day
    return [round(s, 3), round(peak_i, 3), peak_day, round(r, 3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None)],
 [('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('regression: overshoot-prone large beta two steps', (2.5, 0.3, 200, 80, 6, 2), [0.0, 156.8, 1, 166.339]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639])],
 [('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: mild outbreak in 1000[996.036, 1.0, 0, 3.962][85.915, 303.71, 39, 810.961]Failed
regression: fast outbreak daily steps[2.433, 249.994, 10, 497.014][2.433, 249.994, 10, 497.014]Passed
regression: subcritical seed decays[987.537, 10.0, 0, 12.454][984.011, 10.0, 0, 15.167]Failed
regression: overshoot-prone large beta single step[0.0, 75.0, 1, 95.312][0.0, 75.0, 1, 95.312]Passed
control: boundary zero days[97.0, 3.0, 0, 0.0][97.0, 3.0, 0, 0.0]Passed
control: boundary no seed[100.0, 0.0, 0, 0.0][100.0, 0.0, 0, 0.0]Passed
control: invalid seed above populationNoneNonePassed

SHA-256 / 5fbd4abfc8bbd8b37575741dd6d92572cd36ce533e486f648ff20a5f4c4a16b7

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(beta, gamma, pop, i0, days, substeps):
    if pop <= 0 or i0 < 0 or i0 > pop or substeps < 1:
        return None
    dt = 1.0 / substeps
    s = float(pop - i0)
    i = float(i0)
    r = 0.0
    peak_i = i
    peak_day = 0
    for day in range(1, days + 1):
        for _ in range(substeps):
            new_inf = min(beta * s * i / pop * dt, s)
            new_rec = gamma * i0 * dt
            s -= new_inf
            i += new_inf - new_rec
            r += new_rec
        if i > peak_i:
            peak_i = i
            peak_day = day
    return [round(s, 3), round(peak_i, 3), peak_day, round(r, 3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None)],
 [('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('regression: overshoot-prone large beta two steps', (2.5, 0.3, 200, 80, 6, 2), [0.0, 156.8, 1, 166.339]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639])],
 [('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: mild outbreak in 1000[0.021, 994.557, 51, 6.0][85.915, 303.71, 39, 810.961]Failed
regression: fast outbreak daily steps[0.0, 487.812, 12, 40.0][2.433, 249.994, 10, 497.014]Failed
regression: subcritical seed decays[1007.013, 10.0, 0, 40.0][984.011, 10.0, 0, 15.167]Failed
regression: overshoot-prone large beta single step[0.0, 75.0, 1, 125.0][0.0, 75.0, 1, 95.312]Failed
control: boundary zero days[97.0, 3.0, 0, 0.0][97.0, 3.0, 0, 0.0]Passed
control: boundary no seed[100.0, 0.0, 0, 0.0][100.0, 0.0, 0, 0.0]Passed
control: invalid seed above populationNoneNonePassed

SHA-256 / 7b8beed9f06bb4979a44ae398fa269e92a333789a97586e10a2fc666cec51a88

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(beta, gamma, pop, i0, days, substeps):
    if pop <= 0 or i0 < 0 or i0 > pop or substeps < 1:
        return None
    dt = 1.0 / substeps
    s = float(pop - i0)
    i = float(i0)
    r = 0.0
    peak_i = i
    peak_day = 0
    for day in range(1, days + 1):
        for _ in range(substeps):
            new_inf = min(beta * s * i / pop * dt, s)
            new_rec = gamma * i * dt
            s -= new_inf
            i += new_inf - new_rec
            r += new_rec
        if i > peak_i:
            peak_i = i
            peak_day = day
    return [round(s, 3), round(peak_i, 3), peak_day, round(r, 3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None)],
 [('regression: overshoot-prone large beta single step', (3.0, 0.5, 100, 50, 5, 1), [0.0, 75.0, 1, 95.312]),
  ('regression: overshoot-prone large beta two steps', (2.5, 0.3, 200, 80, 6, 2), [0.0, 156.8, 1, 166.339]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639])],
 [('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('control: invalid zero substeps', (0.5, 0.1, 100, 1, 10, 0), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: slow recovery long run', (0.25, 0.05, 2000, 2, 90, 3), [19.331, 965.817, 43, 1829.639]),
  ('regression: large city fine steps',
   (0.4, 0.15, 100000, 20, 50, 8),
   [13105.196, 25867.476, 37, 75829.968]),
  ('regression: small village', (0.6, 0.25, 50, 1, 30, 2), [5.845, 11.78, 12, 43.206]),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])],
 [('regression: mild outbreak in 1000', (0.3, 0.1, 1000, 1, 60, 4), [85.915, 303.71, 39, 810.961]),
  ('regression: fast outbreak daily steps', (0.9, 0.2, 500, 5, 40, 1), [2.433, 249.994, 10, 497.014]),
  ('regression: subcritical seed decays', (0.08, 0.2, 1000, 10, 20, 2), [984.011, 10.0, 0, 15.167]),
  ('control: boundary zero days', (0.5, 0.1, 100, 3, 0, 2), [97.0, 3.0, 0, 0.0]),
  ('control: boundary no seed', (0.5, 0.1, 100, 0, 10, 2), [100.0, 0.0, 0, 0.0]),
  ('control: invalid seed above population', (0.5, 0.1, 100, 150, 10, 2), None),
  ('regression: whole population seeded', (0.5, 0.2, 40, 40, 5, 2), [0.0, 40.0, 0, 26.053])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: mild outbreak in 1000[85.915, 303.71, 39, 810.961][85.915, 303.71, 39, 810.961]Passed
regression: fast outbreak daily steps[2.433, 249.994, 10, 497.014][2.433, 249.994, 10, 497.014]Passed
regression: subcritical seed decays[984.011, 10.0, 0, 15.167][984.011, 10.0, 0, 15.167]Passed
regression: overshoot-prone large beta single step[0.0, 75.0, 1, 95.312][0.0, 75.0, 1, 95.312]Passed
control: boundary zero days[97.0, 3.0, 0, 0.0][97.0, 3.0, 0, 0.0]Passed
control: boundary no seed[100.0, 0.0, 0, 0.0][100.0, 0.0, 0, 0.0]Passed
control: invalid seed above populationNoneNonePassed

SHA-256 / 2e1e47e0282579989fa8ef1ea7233739cba9a1d700e8bb9b09ab92a44e91e56c

Verification & scope

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.

Observations recorded using Python 3.12.14 at 2026-09-29T14:47:28.683674+00:00.

Case digest / 1d6afa5dad4b96ad14ac6072de5e4af851276acc77d8f9088e1b465d6b0d4be0