# How Epidemics Spread 9, Exercise 3
# Exercise 3: combining a contact cut with case isolation. R0 = 2.5 (assumed); a contact cut c gives R = 2.5 * (1 - c).
import math
import random

def poisson(lam):
    """Poisson random number (Knuth's method; fine for small lam)."""
    limit, k, p = math.exp(-lam), 0, 1.0
    while True:
        p *= random.random()
        if p <= limit:
            return k
        k += 1

def escaped(R, detect_mean, cap=300, seed_cases=3):
    """Isolation only (no tracing): True if the outbreak reaches `cap` cases."""
    queue = [(0.0, random.gammavariate(4, detect_mean / 4)) for _ in range(seed_cases)]
    total = 0
    while queue:
        t, iso = queue.pop()
        total += 1
        if total >= cap:
            return True
        for _ in range(poisson(R)):
            child_t = t + random.gammavariate(4, 1.25)
            if child_t < iso:
                queue.append((child_t, child_t + random.gammavariate(4, detect_mean / 4)))
    return False

random.seed(33)
runs = 200
print("contact cut   R after cut   escape % (detect 7 d)   (detect 5 d)   (detect 3 d)")
for c in (0.0, 0.2, 0.4, 0.5):
    R = 2.5 * (1 - c)
    row = [sum(escaped(R, d) for _ in range(runs)) / runs * 100 for d in (7, 5, 3)]
    print(f"{c*100:9.0f}% {R:13.2f} {row[0]:21.0f} {row[1]:14.0f} {row[2]:14.0f}")
# Output when run:
# contact cut   R after cut   escape % (detect 7 d)   (detect 5 d)   (detect 3 d)
#         0%          2.50                    96             68              0
#        20%          2.00                    78             12              0
#        40%          1.50                    19              0              0
#        50%          1.25                     0              0              0

# Reading the output: a contact cut and quicker isolation work together because both lower the number of people each case infects. With no contact cut, detection after 7 days leaves 96 per cent of outbreaks escaping, 5 days 68 per cent, and 3 days none in these 200 runs. A 40 per cent cut with 7-day detection brings escape down to 19 per cent, and a 50 per cent cut stops it at 0. With 3-day detection nothing escapes even without a cut, in this model. The model is optimistic: every case is detected and isolation is perfect. The values are assumed.