Chapter 6 — Calculus II

Chapter 6 — Calculus II

Lesson map

Lesson 06 published. Map, example, EX016.

flowchart TD
  N0["Learning outcomes"]
  N1["6.1 Modelling decay"]
  N2["6.2 Comparing schedules"]
  N3["Lab — Average benefit of annual versus quarterly ref"]
  N4["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4
  N4 --> EX["Worked example"]
  EX --> Q["Exercise EX016"]

Logical drawing

flowchart LR
  N0["Learning outcomes"]
  N1["6.1 Modelling decay"]
  N2["6.2 Comparing schedules"]
  N3["Lab — Average benefit of annual versus quarterly ref"]
  N4["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX016"]

[Learning outcomes]
    |
    v
[6.1 Modelling decay]
    |
    v
[6.2 Comparing schedules]
    |
    v
[Lab — Average benefit of annual versus quarterly ref]
    |
    v
[Synthesis]
    |
    v
[Worked example] --> [Exercise EX016]

Learning outcomes

You will (1) model decay of training effects; (2) compare refresher schedules; (3) recognise the limits of smooth models of behaviour.

6.1 Modelling decay

Aiko reads that skills and attention fade after training. She models the share of staff who would correctly spot a typical lure as starting high after training and decaying exponentially towards a lower baseline. The model has two parameters: the baseline and the rate of decay. Calculus lets her compute the half-life, the time for the extra benefit of training to halve. She emphasises to Prof. Kerr that this is a model to support planning, and that she has no evidence yet that Greyford's staff follow this exact curve.

6.2 Comparing schedules

Using the model, Aiko compares a single annual course with short refreshers every three months. Integrating the benefit over a year shows that frequent short refreshers keep the average higher, even if each is less intense. The lab computes the average benefit for both schedules using the same decay rate. She notes that the comparison ignores the time each schedule costs staff, which the compliance budget idea of Beautement, Sasse and Wonham (2008) says must be counted, so she adds staff minutes to the comparison.

Lab — Average benefit of annual versus quarterly refreshers

# Average training benefit over 12 months: annual vs quarterly refreshers (synthetic exponential decay).
import math
base, gain, k = 0.40, 0.45, 0.24          # baseline, boost after training, decay per month
print(f"half-life of benefit: {math.log(2) / k:.1f} months")
def avg(refresh_months):
    total, steps = 0.0, 1200
    for i in range(steps):
        t = 12 * i / steps; since = min(t - r for r in refresh_months if r <= t)
        total += base + gain * math.exp(-k * since)
    return total / steps
print(f"annual (month 0): average {avg([0]):.2f}")
print(f"quarterly (0,3,6,9): average {avg([0, 3, 6, 9]):.2f}")

Run with python3 lab_decay.py. Verified output:

half-life of benefit: 2.9 months
annual (month 0): average 0.55
quarterly (0,3,6,9): average 0.72

6.3 Limits of smooth models

Real behaviour is lumpy. A single memorable incident can raise vigilance more than any refresher, and a busy period can lower it suddenly. Bada, Sasse and Nurse (2019) found that awareness campaigns often fail when they rely on information alone, so a smooth decay curve may describe knowledge rather than behaviour. Aiko writes these limits beside her model and proposes testing it against Greyford's actual reporting data before using it to set the training calendar. She notes this test as a task for her later measurement chapter, where real data will be available.

Worked example

Aiko models training decay with a half-life, compares annual and quarterly schedules including staff time, and records the model's limits.

Synthetic case. Parameters are synthetic; Beautement et al. (2008) and Bada et al. (2019) are real.

Errors and limits

Exponential decay is an assumption. Knowledge measures may not reflect behaviour.

Exercises

• EX016 — Half-life. What half-life does the lab's decay rate imply? Rubric: 3 = about 2.9 months;; 2 = about 3 months;; 1 = vague;; 0 = none.
• EX017 — Schedule result. Which schedule kept the higher average benefit in the lab? Rubric: 3 = quarterly: 0.72 versus 0.55 for annual;; 2 = quarterly;; 1 = annual;; 0 = none.
• EX018 — Missing cost. What cost did Aiko add to the comparison? Rubric: 3 = staff minutes spent on training;; 2 = staff time;; 1 = vague;; 0 = none.

Synthesis

Calculus models of decay help compare schedules, provided their assumptions and costs are stated.

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