How Does a Cold Spread Through a School?

One school at a time

The number you choose is called R: how many others each sick student passes the cold to, on average, while nearly everyone is still healthy. Here a sick student has 4 meetings a day, with students picked at random, for 3 days. Each meeting passes the cold on with a small chance, set so that the 12 meetings make R new cases on average.

• a.Choose 0.5 and play five schools. How many catch it, at most?
• b.Choose 2 and play five schools. Does the cold ever die out early?
• c.Choose 1.2. What happens? (Play it several times: it is the hardest to guess.)

Stay home!

A sick student who stays home passes the cold to nobody at school. In the widget, tap a red student to send them home. (A student who caught it today doesn’t feel ill yet: they spend one day at school first, and can be sent home after that.)

Choose 2. Press Next day, and send home every sick student you can. Can you stop the cold before it reaches 10 students?

A hundred schools

One school can be lucky or unlucky. To see what usually happens, run many.

Rounds of passing it on

Think of the cold passing on in rounds. The first student passes it to R others, on average: round 1. Each of those passes it to R more, so round 2 has R × R new cases, and so on. Each round, the new cases are multiplied by R.

(These are averages: over 100 schools, round 2 would have about 64 cases in all.)

Multiply again and again by a number below 1, and the rounds shrink to nothing; by a number above 1, and they grow, faster each round. That is the tipping point you saw in the hundred schools.

Why it stops before everyone

With R = 1.5 the table grows for ever, but a school has only 300 students. As more have had the cold, more meetings are with someone who can’t catch it again. Once half the school has had it, each sick student passes it to about 1.5 × ½ = 0.75 others, below 1. The cold grows until about a third has had it (1.5 × ⅔ = 1), then shrinks, and many never catch it.

With blocks: the rounds

Here are the rounds as a block program: it asks for R, then 10 times multiplies the new cases by R and adds them to the total.

In Python (from Grade 8)

Step 1: one school

The school is two lists, one place for each student, 0 to 299: days_left[s], how many more days student s will be sick (0 if not sick), and caught[s], whether s has ever had the cold. [0] * 300 makes a list of 300 zeros.

From the random module: random.randint(0, 299) picks a student, and random.random() gives a decimal between 0 and 1, so random.random() < CHANCE is true with exactly that chance. random.seed(5) makes the same random numbers come out every time, so your school matches the one here; change the 5 for another.

Today’s new cases only start passing it on tomorrow, so they go into a list new first. The program prints a bar for each day, one # for every two students sick. (days_left.count(0) counts the 0s; str(day).rjust(2) writes the day in two places, so the bars line up.)

import randomrandom.seed(5) STUDENTS = 300      # students in the schoolMEETS = 4           # students a sick student meets each dayDAYS_SICK = 3       # days a student can pass the cold onR = 1.5             # how many each sick student infects, on averageCHANCE = R / (MEETS * DAYS_SICK) days_left = [0] * STUDENTS      # days each student is still sick (0: not sick)caught = [False] * STUDENTS     # has each student had the cold?days_left[0] = DAYS_SICK        # student 0 comes to school with a coldcaught[0] = True day = 0while days_left.count(0) < STUDENTS:      # while anyone is sick    day = day + 1    new = []    for s in range(STUDENTS):        if days_left[s] > 0:              # s is sick today            for m in range(MEETS):                if random.random() < CHANCE:     # this meeting passes it on                    other = random.randint(0, STUDENTS - 1)                    if not caught[other]:                        caught[other] = True                        new.append(other)            days_left[s] = days_left[s] - 1    for s in new:                         # today's new cases start tomorrow        days_left[s] = DAYS_SICK    sick = STUDENTS - days_left.count(0)    print(str(day).rjust(2), '#' * (sick // 2), sick) print('Caught it:', caught.count(True), 'of', STUDENTS)

 1  1 2 # 2 3 ## 5 4 ### 7 5 ##### 10 6 ###### 12 7 ####### 14 8 ####### 15 9 ####### 1410 ######## 1711 ########## 2112 ############# 2713 ############### 3014 ################## 3715 ###################### 4416 ######################## 4917 ######################## 4818 ##################### 4319 #################### 4020 ############## 2921 ########### 2322 ######## 1623 ####### 1424 #### 825 # 326 # 227 # 228 # 229  130  0Caught it: 179 of 300

The bars are the shape of an outbreak: slow at first, then a hump, then a tail. Try R = 0.8, and other seeds.

Step 2: the school as a film

Now the same outbreak as a film, a picture a day, played by show_frames from the page’s own pixels module (on your own computer you might use Pillow). picture() turns the lists into 15 rows of 20 seats: 0 healthy, 1 sick, 2 better.

import randomfrom pixels import show_framesrandom.seed(5) STUDENTS = 300MEETS = 4DAYS_SICK = 3R = 1.5CHANCE = R / (MEETS * DAYS_SICK) days_left = [0] * STUDENTScaught = [False] * STUDENTSdays_left[0] = DAYS_SICKcaught[0] = True def picture():    rows = []    for r in range(15):        row = []        for c in range(20):            s = r * 20 + c                # the student in row r, seat c            if days_left[s] > 0:                row.append(1)             # sick            elif caught[s]:                row.append(2)             # better            else:                row.append(0)             # healthy        rows.append(row)    return rows frames = [picture()]day = 0while days_left.count(0) < STUDENTS:    day = day + 1    new = []    for s in range(STUDENTS):        if days_left[s] > 0:            for m in range(MEETS):                if random.random() < CHANCE:     # this meeting passes it on                    other = random.randint(0, STUDENTS - 1)                    if not caught[other]:                        caught[other] = True                        new.append(other)            days_left[s] = days_left[s] - 1    for s in new:        days_left[s] = DAYS_SICK    frames.append(picture()) print('The cold lasted', day, 'days.')show_frames(frames, ['#f1ead0', '#d6332a', '#8fa3b8'])

The cold lasted 30 days.

Picture

Press Play under the picture, or move the slider.

Step 3: a hundred schools for each R

Step 1’s loop goes into a function, one_school(R), which gives back how many caught the cold, and the program runs a hundred schools for each R. Fewer than 30 cases counts as died out early.

import randomrandom.seed(2026) STUDENTS = 300MEETS = 4DAYS_SICK = 3 def one_school(R):    chance = R / (MEETS * DAYS_SICK)    days_left = [0] * STUDENTS    caught = [False] * STUDENTS    days_left[0] = DAYS_SICK    caught[0] = True    while days_left.count(0) < STUDENTS:        new = []        for s in range(STUDENTS):            if days_left[s] > 0:                for m in range(MEETS):                    if random.random() < chance:     # this meeting passes it on                        other = random.randint(0, STUDENTS - 1)                        if not caught[other]:                            caught[other] = True                            new.append(other)                days_left[s] = days_left[s] - 1        for s in new:            days_left[s] = DAYS_SICK    return caught.count(True) for R in [0.5, 0.8, 0.9, 1.1, 1.2, 1.5, 2, 3]:    total = 0    early = 0    for school in range(100):        n = one_school(R)        total = total + n        if n < 30:            early = early + 1    print('R', R, ' average caught:', total / 100, ' died out early:', early, 'of 100')

R 0.5  average caught: 1.99  died out early: 100 of 100R 0.8  average caught: 4.26  died out early: 99 of 100R 0.9  average caught: 6.22  died out early: 95 of 100R 1.1  average caught: 19.88  died out early: 79 of 100R 1.2  average caught: 27.25  died out early: 74 of 100R 1.5  average caught: 105.17  died out early: 38 of 100R 2  average caught: 190.15  died out early: 21 of 100R 3  average caught: 270.18  died out early: 4 of 100

Step 4: sick students stay home

In this model a student feels ill from their second day. Suppose every sick student stays home from then on: they meet people only on their first day, when days_left[s] is still DAYS_SICK. One new if does it.

import randomrandom.seed(2026) STUDENTS = 300MEETS = 4DAYS_SICK = 3 def one_school(R, stay_home):    chance = R / (MEETS * DAYS_SICK)    days_left = [0] * STUDENTS    caught = [False] * STUDENTS    days_left[0] = DAYS_SICK    caught[0] = True    while days_left.count(0) < STUDENTS:        new = []        for s in range(STUDENTS):            if days_left[s] > 0:                first_day = days_left[s] == DAYS_SICK                if first_day or not stay_home:                    for m in range(MEETS):                        if random.random() < chance:     # this meeting passes it on                            other = random.randint(0, STUDENTS - 1)                            if not caught[other]:                                caught[other] = True                                new.append(other)                days_left[s] = days_left[s] - 1        for s in new:            days_left[s] = DAYS_SICK    return caught.count(True) for R in [1.5, 2, 3]:    for stay_home in [False, True]:        total = 0        for school in range(100):            total = total + one_school(R, stay_home)        print('R', R, ' stay home:', stay_home, ' average caught:', total / 100)

R 1.5  stay home: False  average caught: 107.78R 1.5  stay home: True  average caught: 1.92R 2  stay home: False  average caught: 201.79R 2  stay home: True  average caught: 3.17R 3  stay home: False  average caught: 262.0R 3  stay home: True  average caught: 10.66

• Below 1, a cold fizzles out. With R = 0.5 every school had fewer than 30 cases, and with 0.8 all but one, with 4.26 cases on average: close to the blocks’ 5.
• Just above 1, it’s a gamble. At R = 1.1 and 1.2 most schools were lucky (79 and 74 of 100); the rest had big outbreaks.
• Well above 1, it reaches most of the school: 105 on average at R = 1.5, 190 at 2, 270 at 3. A change from 0.8 to 1.5 turned a handful into a third of the school.
• Changing what people do changes R. With sick students staying home after their first day, R = 3 became about 1: 11 cases on average instead of 262.

That is why health workers talk so much about R. They can’t change the virus, but they can change how many people each sick person meets. Push R below 1 and the outbreak dies out by itself: you needn’t stop every case, only enough of them.

Our school is much simpler than a real one:

• Everyone meets everyone at random. In a real school you meet your own class and your friends far more than anyone else, so colds spread class by class.
• Everyone is the same. Real students differ: some meet many people, some few.
• The numbers are made up. 4 meetings a day and 3 days sick keep the program small. The NHS (Britain’s health service) says you can pass on a cold until all your symptoms have gone, usually 1 to 2 weeks, and from a few days before you feel ill, so staying home helps but can’t stop everything.
• No homes, weekends or teachers, and one cold each. Real colds travel home and back, and you can catch another soon after: the CDC (the United States’ health agency) says more than 200 viruses cause colds.

A model like this shows the shape of what happens, and why: the tipping point at 1, the luck at the start, the cold stopping before everyone has it. For real numbers, scientists check models against real cases, as Grade 11’s Modelling and Simulation does.

Half stay home. In Step 4, give each sick student a chance, say random.random() < 0.5, of staying home. How many must stay home to bring R = 2 below 1?

A vaccine. Before the first day, mark some students caught = True. With R = 3, how many in 100 must be immune for the cold to die out early in most schools? Compare “Your turn” above.

★ Enough for a whole school (blocks)

Finish the program so that it says how many rounds it takes before the total reaches at least 300, a whole school. Say just the number (for R = 2, it says 8). Then press Test.

★★ Who passed it on most? (Python, from Grade 8)

The school nurse keeps a notebook: for every new case, the day, the student who passed the cold on, and the student who caught it, as [day, giver, taker]. The first student is 0, and the students of this class are numbered 0 to 39. Write top_spreader(log) that gives back [student, count]: the student who passed it to the most others, and how many. If two passed it on equally often, give the one with the smaller number.

def top_spreader(log):    # log is a list of [day, giver, taker]    return [0, 0] notebook = [[1, 0, 7], [1, 0, 12], [2, 7, 3], [3, 12, 5], [3, 12, 9], [4, 9, 2]]print(top_spreader(notebook))

★★★ Generation by generation (Python, from Grade 8)

Student 0 is generation 0. The students who caught it from student 0 are generation 1, those who caught it from generation 1 are generation 2, and so on. Write generations(log) that gives back a list of how many students are in each generation: for the notebook below, [1, 2, 3, 1]. The log is in order of days, so a giver always appears as a taker (or is student 0) before they give it to anyone.

def generations(log):    # log is a list of [day, giver, taker], in order of days    return [1] notebook = [[1, 0, 7], [1, 0, 12], [2, 7, 3], [3, 12, 5], [3, 12, 9], [4, 9, 2]]print(generations(notebook))

• Grade 7, Variables and Choices: a variable that grows each time round a loop.
• Grade 8, Decisions and Loops in Python: the for and while loops this school runs on.
• Grade 8, Data, Privacy and You: who should see a school’s list of who was sick?
• Grade 11, Modelling and Simulation: an epidemic in a whole district, and checking a model against real data.
• How Far Does a Wandering Ant Get? Chance, step by step, gives a surprising answer again.
• How can throwing rice find π? Thousands of random throws give an exact number.