FAILURE MAP
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FA-64916 / Epidemic compartment models / Open access

SEIR R0 with vital dynamics and vaccine coverage: efficacy adjustment · case 01

Imperfect vaccines appear to need lower coverage than perfect ones.

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

ROOT CAUSE

Coverage multiplies by efficacy instead of dividing by it.

VERIFIED REPAIR

Restore the efficacy adjustment rule: `coverage = threshold / efficacy`.

Unsuccessful approach: Adding the failure fraction is not the all-or-nothing coverage adjustment.

Case contract

R0 = beta*sigma/((sigma+mu)*(gamma+mu)); when R0<=1 return [R0,0.0,0.0]; herd threshold 1-1/R0; critical coverage threshold/efficacy, None if it exceeds 1 (exactly 1 is feasible); values rounded to 6; None for invalid rates or efficacy outside (0,1].

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, sigma, gamma, mu, efficacy):
    if min(beta, sigma, gamma) <= 0 or mu < 0 or not 0 < efficacy <= 1:
        return None
    r0 = beta * sigma / ((sigma + mu) * (gamma + mu))
    if r0 <= 1:
        return [round(r0, 6), 0.0, 0.0]
    threshold = 1 - 1 / r0
    coverage = threshold * efficacy
    if coverage > 1:
        coverage = None
    else:
        coverage = round(coverage, 6)
    return [round(r0, 6), round(threshold, 6), coverage]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6])],
 [('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)]]
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: measles-like high R0[14.4, 0.930556, 0.884028][14.4, 0.930556, 0.979532]Failed
regression: flu-like moderate R0 with births[1.780627, 0.4384, 0.26304][1.780627, 0.4384, 0.730667]Failed
regression: exact R0 of two with half efficacy[2.0, 0.5, 0.25][2.0, 0.5, 1.0]Failed
control: subcritical with mortality[0.8, 0.0, 0.0][0.8, 0.0, 0.0]Passed
control: boundary R0 exactly one[1.0, 0.0, 0.0][1.0, 0.0, 0.0]Passed
regression: infeasible coverage poor vaccine[5.529954, 0.819167, 0.409583][5.529954, 0.819167, None]Failed
control: high mortality relative to latency[2.5, 0.6, 0.6][2.5, 0.6, 0.6]Passed

SHA-256 / fc64e728ac27e44b6efe6cf3718e96568386ad7811255f578ad1601e66e05927

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(beta, sigma, gamma, mu, efficacy):
    if min(beta, sigma, gamma) <= 0 or mu < 0 or not 0 < efficacy <= 1:
        return None
    r0 = beta * sigma / ((sigma + mu) * (gamma + mu))
    if r0 <= 1:
        return [round(r0, 6), 0.0, 0.0]
    threshold = 1 - 1 / r0
    coverage = threshold + (1 - efficacy)
    if coverage > 1:
        coverage = None
    else:
        coverage = round(coverage, 6)
    return [round(r0, 6), round(threshold, 6), coverage]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6])],
 [('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)]]
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: measles-like high R0[14.4, 0.930556, 0.980556][14.4, 0.930556, 0.979532]Failed
regression: flu-like moderate R0 with births[1.780627, 0.4384, 0.8384][1.780627, 0.4384, 0.730667]Failed
regression: exact R0 of two with half efficacy[2.0, 0.5, 1.0][2.0, 0.5, 1.0]Passed
control: subcritical with mortality[0.8, 0.0, 0.0][0.8, 0.0, 0.0]Passed
control: boundary R0 exactly one[1.0, 0.0, 0.0][1.0, 0.0, 0.0]Passed
regression: infeasible coverage poor vaccine[5.529954, 0.819167, None][5.529954, 0.819167, None]Passed
control: high mortality relative to latency[2.5, 0.6, 0.6][2.5, 0.6, 0.6]Passed

SHA-256 / 0bbad1f473f1d49627f06191e16db0fde37cb40b1be315cdbb7a1dfcc6e77b16

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(beta, sigma, gamma, mu, efficacy):
    if min(beta, sigma, gamma) <= 0 or mu < 0 or not 0 < efficacy <= 1:
        return None
    r0 = beta * sigma / ((sigma + mu) * (gamma + mu))
    if r0 <= 1:
        return [round(r0, 6), 0.0, 0.0]
    threshold = 1 - 1 / r0
    coverage = threshold / efficacy
    if coverage > 1:
        coverage = None
    else:
        coverage = round(coverage, 6)
    return [round(r0, 6), round(threshold, 6), coverage]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6])],
 [('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None),
  ('control: invalid negative mortality', (1.0, 0.5, 0.2, -0.1, 0.9), None),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: measles-like high R0', (1.8, 0.1, 0.125, 0.0, 0.95), [14.4, 0.930556, 0.979532]),
  ('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('control: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('regression: flu-like moderate R0 with births', (0.5, 0.5, 0.25, 0.02, 0.6), [1.780627, 0.4384, 0.730667]),
  ('regression: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('control: subcritical with mortality', (0.3, 0.2, 0.25, 0.05, 0.9), [0.8, 0.0, 0.0]),
  ('control: boundary R0 exactly one', (1.0, 1.0, 1.0, 0.0, 0.8), [1.0, 0.0, 0.0]),
  ('regression: infeasible coverage poor vaccine', (1.2, 0.3, 0.2, 0.01, 0.5), [5.529954, 0.819167, None]),
  ('control: high mortality relative to latency', (3.0, 0.1, 0.5, 0.1, 1.0), [2.5, 0.6, 0.6]),
  ('control: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)]]
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: measles-like high R0[14.4, 0.930556, 0.979532][14.4, 0.930556, 0.979532]Passed
regression: flu-like moderate R0 with births[1.780627, 0.4384, 0.730667][1.780627, 0.4384, 0.730667]Passed
regression: exact R0 of two with half efficacy[2.0, 0.5, 1.0][2.0, 0.5, 1.0]Passed
control: subcritical with mortality[0.8, 0.0, 0.0][0.8, 0.0, 0.0]Passed
control: boundary R0 exactly one[1.0, 0.0, 0.0][1.0, 0.0, 0.0]Passed
regression: infeasible coverage poor vaccine[5.529954, 0.819167, None][5.529954, 0.819167, None]Passed
control: high mortality relative to latency[2.5, 0.6, 0.6][2.5, 0.6, 0.6]Passed

SHA-256 / 4f349110a3c4254caadf5c2110ca95ae8900fbbf22bfb841be8c69cd7eafe066

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 2026-09-29T14:47:29.216689+00:00.

Case digest / 9f3145d6132e20e9c8b20ebec626eb4ea09baa0a09cd0a8de3726db23286846f