FA-64896 / Epidemic compartment models / Open access

SEIR R0 with vital dynamics and vaccine coverage: latent survival factor · case 01

R0 is overstated for pathogens with long latency in short-lived hosts.

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

ROOT CAUSE

The probability sigma/(sigma+mu) of surviving the latent stage is omitted.

VERIFIED REPAIR

Restore the latent survival factor rule: `beta * sigma / ((sigma + mu) * (gamma + mu))`.

Unsuccessful approach: Subtracting mu in the latent factor inflates survival instead of discounting it.

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 / (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 = [[('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)],
 [('regression: 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: 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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('control: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: 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: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773])]]
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 fixture	Actual	Expected	Outcome
control: measles-like high R0	`[14.4, 0.930556, 0.979532]`	`[14.4, 0.930556, 0.979532]`	Passed
regression: flu-like moderate R0 with births	`[1.851852, 0.46, 0.766667]`	`[1.780627, 0.4384, 0.730667]`	Failed
control: exact R0 of two with half efficacy	`[2.0, 0.5, 1.0]`	`[2.0, 0.5, 1.0]`	Passed
regression: subcritical with mortality	`[1.0, 0.0, 0.0]`	`[0.8, 0.0, 0.0]`	Failed
control: boundary R0 exactly one	`[1.0, 0.0, 0.0]`	`[1.0, 0.0, 0.0]`	Passed
regression: infeasible coverage poor vaccine	`[5.714286, 0.825, None]`	`[5.529954, 0.819167, None]`	Failed
control: invalid efficacy zero	`None`	`None`	Passed

SHA-256 / fba21ed2d4cb9510a20324fd9926c0c0b4172e3ca711bf384441348d6757dcc8

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 / 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 = [[('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)],
 [('regression: 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: 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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('control: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: 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: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773])]]
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 fixture	Actual	Expected	Outcome
control: measles-like high R0	`[14.4, 0.930556, 0.979532]`	`[14.4, 0.930556, 0.979532]`	Passed
regression: flu-like moderate R0 with births	`[1.929012, 0.4816, 0.802667]`	`[1.780627, 0.4384, 0.730667]`	Failed
control: exact R0 of two with half efficacy	`[2.0, 0.5, 1.0]`	`[2.0, 0.5, 1.0]`	Passed
regression: subcritical with mortality	`[1.333333, 0.25, 0.277778]`	`[0.8, 0.0, 0.0]`	Failed
control: boundary R0 exactly one	`[1.0, 0.0, 0.0]`	`[1.0, 0.0, 0.0]`	Passed
regression: infeasible coverage poor vaccine	`[5.91133, 0.830833, None]`	`[5.529954, 0.819167, None]`	Failed
control: invalid efficacy zero	`None`	`None`	Passed

SHA-256 / c199425505dee2a8bfe27617200586f702e5ef1c019743c78eb764a9b2954410

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 = [[('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: invalid efficacy zero', (1.0, 0.5, 0.2, 0.0, 0.0), None)],
 [('regression: 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: 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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('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: 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])],
 [('control: 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: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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]),
  ('regression: long latency high mortality', (2.4, 0.05, 0.3, 0.04, 0.85), [3.921569, 0.745, 0.876471])],
 [('control: exact R0 of two with half efficacy', (2.0, 1.0, 1.0, 0.0, 0.5), [2.0, 0.5, 1.0]),
  ('regression: 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: 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: perfect vaccine', (0.9, 0.25, 0.2, 0.001, 1.0), [4.459773, 0.775773, 0.775773])]]
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 fixture	Actual	Expected	Outcome
control: 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
control: exact R0 of two with half efficacy	`[2.0, 0.5, 1.0]`	`[2.0, 0.5, 1.0]`	Passed
regression: 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: invalid efficacy zero	`None`	`None`	Passed

SHA-256 / e7a87e8c8a2e5124f88477afea19f525cb20ebefb9299a22910e45f12f4154d9

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 2026-09-29T14:47:28.885710+00:00.

Case digest / 6a9cf2fc90ebb555797345ecba9e3364963890ecaeee36fbbf8acfe44188873e