Controlling Spread

How Epidemics Spread

Chapter 9 · Controlling Spread

On 27 July 1377 the Great Council of Ragusa, today's Dubrovnik, passed a law that kept travellers from plague-stricken places out of the city until they had waited a month on the islands offshore. The waiting period was first thirty days (a trentino) and was later lengthened to forty (a quarantena, from which we get “quarantine”). In 1423 Venice put a hospital for plague victims on the island of Santa Maria di Nazareth, an early lazaretto. Nobody then knew about germs, and the logic was simply that sick people spread sickness and keeping people apart for long enough should stop it. Six hundred years later, the logic is the same, but we can ask sharper questions. Which separation matters most: of the sick, of the possibly exposed, or of everyone? How quickly must it happen? And what happens when we stop? This chapter answers those with two small simulations: a model of a closure that cuts contacts, and a model of case isolation plus contact tracing.

What was run, and what to trust

Every Python snippet in this chapter and in the three exercise solutions was executed, and the output in the comments is what Python printed (random seeds are fixed, so you should get the same numbers with the same Python random generator; the closure model uses no random numbers). The history comes from search-result summaries, which are often truncated or model-written, so check the originals before quoting. The sources agreed on 27 July 1377 as the date of the Ragusa law, on thirty days at first and forty later, and on 1423 for the Venice lazaretto, but they conflicted on details: one account gave thirty days for ships and forty for land travellers, another gave thirty days first and forty after a change in 1397, and one source wrongly placed Ragusa under Venice (Ragusa was an independent republic). I have not resolved these. The summary of Hellewell and colleagues, “Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts”, The Lancet Global Health 8 (2020), gave their design (a stochastic model, varying R0, the delay to isolation, the share of contacts traced and the amount of transmission before symptoms) and the broad finding that the more infectious the disease, the more contacts must be traced; I did not see the full text. The simulations here are my own simplified models, not a reproduction of theirs. Every population, R0, delay, tracing share and contact cut is assumed or illustrative, and each result is a property of the model, not a forecast. John Snow removing the Broad Street pump handle in 1854 is general knowledge I did not check this session. No calculus is used: the closure model steps forward in small time steps, as in How Epidemics Spread 3.

Four Kinds of Control

The words are used loosely in the news, so fix them. Isolation separates people known to be infected from everyone else. Quarantine separates people who may have been exposed but are not known to be infected, in case they are. Contact tracing is the work of finding those people: ask each case who they met, then get in touch. Closures and distancing cut contacts for everyone, whether or not they have been exposed. Vaccination, the other great control, was the subject of How Epidemics Spread 4. All of these work on the same two things: they lower the number of people each case infects (R), or they shorten the time a case spends spreading. John Snow’s pump handle is a different kind again: it removes a shared source. The table compares the four main ones.

	Isolation	Quarantine	Contact tracing	Closures and distancing
Who is affected	known cases	people possibly exposed	contacts of known cases	everyone
Needs	testing or symptoms to find cases	a list of who was exposed	interviews, staff, speed	no information, only compliance
Cost per person	high, but few people	moderate, some are not infected	staff time	low each, enormous in total
Weak point	transmission before cases are found	missed contacts, long waits	delays and unknown contacts	the epidemic returns when it ends
Model below	detection delay	traced share and delay	traced share and delay	contact cut c over a window

Closures: When and How Long

A closure lowers the contact rate for a while. In the model that is one multiplication: for the days the closure is in force, the transmission rate is multiplied by (1 − c), where c is the share of contacts cut. The effective reproduction number becomes R0 × (1 − c) × (share still susceptible). The code uses the S-I-R model of How Epidemics Spread 3 with an assumed R0 of 2.5 and an assumed infectious period of 5 days, in a population of 1,000,000 with 10 people infected at the start. It steps in quarter-day steps, and reports three things for six scenarios: the peak share of the population infected at one time, the day of that peak, and the share who are ever infected.

# A closure that cuts contacts, started early or late, lasting a short or a long time. All numbers assumed.
def run_sir(R0, c, start, stop, days=400, n=1_000_000, i0=10, infectious_days=5.0, dt=0.25):
    """Step a simple S-I-R model forward in small time steps.
    The contact factor is (1 - c) from day `start` up to (not including) day `stop`, otherwise 1."""
    gamma = 1.0 / infectious_days
    beta = R0 * gamma
    s, i, r = n - i0, float(i0), 0.0
    peak, peak_day = i, 0.0
    t = 0.0
    while t < days:
        factor = (1 - c) if start <= t < stop else 1.0
        new_inf = beta * factor * s * i / n * dt
        new_rec = gamma * i * dt
        s -= new_inf
        i += new_inf - new_rec
        r += new_rec
        t += dt
        if i > peak:
            peak, peak_day = i, t
    return peak / n, peak_day, (n - s) / n

scenarios = [
    ("no control", 0.0, 0, 0),
    ("70% cut, days 20-80", 0.7, 20, 80),
    ("70% cut, days 40-100", 0.7, 40, 100),
    ("70% cut, days 20-200", 0.7, 20, 200),
    ("40% cut, days 20-80", 0.4, 20, 80),
    ("70% cut, from day 20, never lifted", 0.7, 20, 10_000),
]
print(f"{'scenario':32s} {'peak infected':>14s} {'peak day':>9s} {'ever infected':>14s}")
for name, c, a, b in scenarios:
    pk, pd, fin = run_sir(2.5, c, a, b)
    print(f"{name:32s} {pk*100:13.2f}% {pd:9.0f} {fin*100:13.1f}%")
# Output when run:
# scenario                          peak infected  peak day  ever infected
# no control                               23.72%        41          89.6%
# 70% cut, days 20-80                      22.86%       112          89.3%
# 70% cut, days 40-100                     23.47%        40          65.7%
# 70% cut, days 20-200                     22.83%       255          89.2%
# 40% cut, days 20-80                       6.49%        65          78.1%
# 70% cut, from day 20, never lifted          0.32%        20           1.5%
    

With no control, 23.72 per cent of the population are infected at the peak on day 41, and 89.6 per cent are ever infected. A 70 per cent cut from day 20 to day 80 changes almost nothing in the end: the epidemic is pushed back, the peak arrives on day 112 instead of 41, but it is as tall (22.86 per cent) and the total is the same (89.3 per cent). Extending the closure to day 200 delays the peak to day 255 and again the total is 89.2 per cent. This is the central lesson of closures that end: unless something has changed by the time they lift, the susceptible people are still there and the epidemic picks up where it left off. What could change is a vaccine, a treatment, better tests, or time to build a tracing system. A weaker cut makes a different trade. A 40 per cent cut leaves R at 1.5 during the closure, so cases still grow, but more slowly: the peak is only 6.49 per cent, on day 65, and the total 78.1 per cent. Timing matters too. A 70 per cent cut starting on day 40, just when the epidemic peaks, leaves the peak where it was (23.47 per cent, on day 40, the day the cut started) but lowers the total to 65.7 per cent. My reading is that stopping transmission at the peak prevents much of the “overshoot” in which an uncontrolled epidemic keeps infecting people after most are already immune; I have not tested that. The last row, a cut that is never lifted, keeps cases to 1.5 per cent, but that is a different question from the others: nobody can hold a 70 per cent cut for ever, and the model says nothing about what that costs.

One more caution about the 20 to 200 row. A model with continuous amounts never lets the infection die out completely; a tiny fraction of a person remains and grows back. Real epidemics are made of whole people, and as shown in How Epidemics Spread 6, a tiny number of cases can simply vanish by chance, or be reseeded from outside. Treat the late peak as a feature of the model rather than a prediction.

Scenario (R0 2.5 assumed)	Peak infected	Peak day	Ever infected
No control	23.72%	41	89.6%
70% cut, days 20 to 80	22.86%	112	89.3%
70% cut, days 40 to 100	23.47%	40	65.7%
70% cut, days 20 to 200	22.83%	255	89.2%
40% cut, days 20 to 80	6.49%	65	78.1%
70% cut, never lifted	0.32%	20	1.5%

Isolation, Tracing and the Race Against Time

A closure treats everyone. Isolation and tracing treat only the people who matter, but they have to win a race: each case has to be removed before it has infected too many others. The second model is a branching process, as in How Epidemics Spread 6. Each case infects a random number of people (Poisson, mean R = 2.5, assumed), each after a random lag (gamma-shaped, mean 5 days, as in How Epidemics Spread 8). Each case is detected a random time after being infected, with a mean you can set, and isolated at once; a case infects nobody after it is isolated. With contact tracing, each newly infected person is traced with some probability p: they are quarantined a fixed delay after their infector was isolated, if that is sooner than their own detection. The code starts every outbreak with 3 cases and calls the outbreak escaped if it reaches 300 cases (an assumed cut-off; the Hellewell summary gave 5,000 cases or 12 weeks for its definition of control). It runs 300 outbreaks per plan. Every case is eventually detected here, which is optimistic, and isolation is perfect, which is also optimistic.

# Case isolation and contact tracing as a branching process. All parameters assumed.
import math
import random
import statistics

def one_outbreak(R, detect_mean, p_trace, trace_delay, cap=300, seed_cases=3):
    """Return (escaped, total_cases). Each case infects Poisson(R) people, each after a random lag
    (gamma, mean 5 days). A case is isolated at its own detection time (infection time + gamma delay)
    or, if traced, at parent's isolation time + trace_delay, whichever is first. Isolated cases infect
    nobody afterwards."""
    def poisson(lam):
        limit, k, p = math.exp(-lam), 0, 1.0
        while True:
            p *= random.random()
            if p <= limit:
                return k
            k += 1
    queue = []  # (infection_time, isolation_time)
    total = 0
    for _ in range(seed_cases):
        t0 = 0.0
        queue.append((t0, t0 + random.gammavariate(4, detect_mean / 4)))
    while queue:
        t, iso = queue.pop()
        total += 1
        if total >= cap:
            return True, total
        for _ in range(poisson(R)):
            lag = random.gammavariate(4, 1.25)
            if t + lag >= iso:
                continue                      # parent already isolated: no infection
            child_t = t + lag
            child_iso = child_t + random.gammavariate(4, detect_mean / 4)
            if random.random() < p_trace:     # child is traced and quarantined
                child_iso = min(child_iso, max(child_t, iso + trace_delay))
            queue.append((child_t, child_iso))
    return False, total

random.seed(9)
print(f"{'plan':42s} {'escaped':>8s} {'mean size if contained':>24s}")
plans = [
    ("no isolation (detect after 1000 days)", 1000, 0.0, 0),
    ("isolate cases, detection mean 5 days", 5, 0.0, 0),
    ("isolate cases, detection mean 3 days", 3, 0.0, 0),
    ("detect 5 days + trace 50%, 2-day delay", 5, 0.5, 2),
    ("detect 5 days + trace 90%, 2-day delay", 5, 0.9, 2),
    ("detect 5 days + trace 90%, 5-day delay", 5, 0.9, 5),
]
for name, d, p, td in plans:
    runs = [one_outbreak(2.5, d, p, td) for _ in range(300)]
    esc = sum(1 for e, _ in runs if e)
    small = [n for e, n in runs if not e]
    shown = f"{statistics.mean(small):.1f}" if small else "-"
    print(f"{name:42s} {esc / 300 * 100:7.0f}% {shown:>24s}")
# Output when run:
# plan                                        escaped   mean size if contained
# no isolation (detect after 1000 days)          100%                        -
# isolate cases, detection mean 5 days            59%                     12.3
# isolate cases, detection mean 3 days             0%                      7.9
# detect 5 days + trace 50%, 2-day delay          11%                     37.5
# detect 5 days + trace 90%, 2-day delay           0%                     16.3
# detect 5 days + trace 90%, 5-day delay          63%                     14.3
    

Without isolation every outbreak escapes. Isolating cases with a mean detection time of 5 days is not enough: 59 per cent of outbreaks escape, because by the time a case is found it has already passed the infection on, and each of those cases will do the same. Cutting the detection time to 3 days stops all 300 in this run. Detection speed is a threshold effect in this model, not a gentle slope. Tracing with the 5-day detection and a 2-day delay brings escape down to 11 per cent at 50 per cent tracing and to 0 at 90 per cent. But the same 90 per cent tracing with a 5-day delay does no better than isolation alone (63 per cent escape, against 59 per cent, which is within what 300 runs can scatter by, about 3 points either way by a rough binomial estimate of mine). A trace that takes as long as the detection itself does not catch anyone early. The mean size of the outbreaks that were contained is also informative: 37.5 cases for the 50 per cent trace, since a weaker control lets contained outbreaks run longer before they die out. The direction agrees with the summary of Hellewell and colleagues, that faster isolation and more tracing both help, and that more infectious diseases need more of both. That is a qualitative agreement with a summary, nothing more.

Plan (R = 2.5 assumed, 300 runs)	Escaped	Mean size if contained	What it shows
No isolation	100%	none contained	the baseline
Isolate, 5-day detection	59%	12.3	not fast enough alone
Isolate, 3-day detection	0%	7.9	speed alone can be enough
5-day detection, trace 50%, 2-day delay	11%	37.5	tracing helps a lot
5-day detection, trace 90%, 2-day delay	0%	16.3	high coverage, fast
5-day detection, trace 90%, 5-day delay	63%	14.3	coverage without speed fails

Comparing Strategies Honestly

The two models make different promises. The closure model asks how much total harm a measure prevents, in a well-mixed population. The branching model asks whether a small outbreak can be stopped before it grows. Often the right answer is a combination: a short closure while tracing capacity is built, then tracing and isolation to keep numbers down after reopening. Comparing strategies by simulation needs the same discipline as comparing drugs. Change one thing at a time; run many random repeats and report the spread, as in How Epidemics Spread 6; write down every assumed number; and look for the scenario where your favourite plan fails. The most important numbers to vary are the ones you are least sure of: the delay, the share of cases that are ever found, and the share of contacts you can name. The measures are also not independent in real life (people who isolate may also be the ones who distance), and none of the models here counts the cost of a measure, which is what makes the choice hard.

Common mistakes

Treating a temporary closure as a cure when the susceptible people are still there afterwards; reading the late peak of a model that never lets the infection die out as a prediction; judging tracing by the share of contacts reached and ignoring how long it takes; assuming every case is found and every isolation is perfect (both are optimistic in the model here); comparing two plans on one random run instead of many; mixing up quarantine (possibly exposed) with isolation (known infected); judging a measure by cases prevented without asking what it costs; calling the model a reproduction of a published study when it is only a simplified model in the same spirit; and using the illustrative numbers here (R0 of 2.5, a 5-day interval, 300 cases, a 70 per cent cut) as estimates for a real disease.

Exercise 1

Using the closure model (R0 2.5, a 70 per cent contact cut, both assumed), try starting days of 10, 30 and 50 with closure lengths of 30, 60 and 120 days. Print the peak share infected and the share ever infected for each of the nine. When does a closure change the peak, when does it change only the total, and when does it change neither?

📄 View solution

Exercise 2

Using the branching model with R = 2.5 and a 5-day mean detection time (assumed), find the share of 200 outbreaks that escape for tracing coverages of 30, 50, 70 and 90 per cent and tracing delays of 1, 3 and 5 days. Which matters more at low coverage, and what does a 5-day delay do to the benefit of high coverage?

📄 View solution

Exercise 3

Combine a contact cut with isolation (no tracing). Cut contacts by 0, 20, 40 and 50 per cent, so that R becomes 2.5 × (1 − cut), and use mean detection times of 7, 5 and 3 days. For each pair print the share of 200 outbreaks that escape. How much of a contact cut is needed to make slow detection enough, and how optimistic are the model’s assumptions?

📄 View solution

Quick Reference

Isolation: separate known cases; quarantine: separate people who may have been exposed; tracing: find those people
Closure model: transmission multiplied by (1 − c) between a start and an end day; effective R = R0 × (1 − c) × share susceptible
A closure that ends leaves the susceptible people; unless something changes the epidemic returns (peak delayed, total similar)
Timing: started at the peak, a cut lowered the total but not the peak; started early and short, it only delayed the epidemic (in this model)
Branching model: Poisson children, gamma lags, detection delay, isolated cases infect nobody; escaped = reaches the case cap
Race against time: tracing needs coverage and speed; a trace as slow as detection adds little
Compare strategies: many runs, one change at a time, all assumptions stated, include a scenario where the plan fails
Assumes: well-mixed population, perfect isolation, every case eventually detected, illustrative numbers

Coming next

How Epidemics Spread 10 is the capstone, Plan a Response: the history of famous outbreaks, building a small outbreak simulator and judging a control plan.