Answer key · Unit 11

Disease and immunity.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 4

Lesson 1 of 4 · About 25 minutes

Viruses

Pathogens that are not cells: what is inside a virus, how small it is, how it takes over a cell, why antibiotics cannot touch it, and how fast viruses multiply.

Open the lesson

The 7 steps and their answers

Step 1 of 7
Open this step
Not a cell
Practice · Choose
Question
Both bacteria and viruses cause disease. What is the main difference between them?

Correct answer
- Correct A bacterium is a living cell that divides by itself; a virus is genes in a coat, and copies itself only inside another cell
Setup, choices and feedback
The Scene at this step
Virus
Coronavirus
Infection
0 %
Cut open
Cut open
Choices
Shown in a random order.
- Correct A bacterium is a living cell that divides by itself; a virus is genes in a coat, and copies itself only inside another cell
- Wrong A virus is a very small bacterium
  Misconception: Thinks a virus is just a very small bacterium.
  Feedback from the catalogue
  A bacterium is a living cell: it has a membrane, ribosomes, machinery to release energy, and it divides by itself. A virus is not a cell at all: a genome of DNA or RNA in a protein coat, with no machinery of its own. It is also about twenty times smaller than a typical bacterium. A virus can copy itself only inside someone else’s cell.
- Wrong Bacteria are always harmful, and viruses only sometimes
  Feedback
  Most bacteria are harmless, and many are helpful: trillions live in our gut. The difference is in structure, not in harm.
Feedback for a right answer
Right. Inside the virus there is only a genome. There are no ribosomes, no cytoplasm and no machinery to release energy, which is why biologists still argue over whether to call it alive.

Hint
Cut open at Cut open: what is inside the virus, and what is missing?

Worked solution
1. A bacterium is a whole cell: membrane, cytoplasm, ribosomes and DNA, and it splits in two by itself.
2. A virus is a genome, DNA or RNA, in a protein capsid, sometimes with a fatty envelope.
3. It has no machinery at all, so it can copy itself only inside a cell.
If you chose “a small bacterium”, you thought the difference was size. The difference is that a virus is not a cell at all.
Step 2 of 7
Open this step
How small
Practice · Calculate
Question
A coronavirus is about 100 nm across, and an E. coli bacterium about 2 µm long. How many coronaviruses in a row would match the length of one bacterium?

Correct answer
20 ± 0.5
Setup, choices and feedback
The Scene at this step
Virus
Coronavirus
Infection
0 %
Cut open
Whole
Accepted within
± 0.5
A fixed value in the question
Wrong answers it recognises
- 0.02 ± 0.005
  Misconception: Thinks a virus is just a very small bacterium.
  Feedback from the catalogue
  A bacterium is a living cell: it has a membrane, ribosomes, machinery to release energy, and it divides by itself. A virus is not a cell at all: a genome of DNA or RNA in a protein coat, with no machinery of its own. It is also about twenty times smaller than a typical bacterium. A virus can copy itself only inside someone else’s cell.
Feedback for a right answer
$\dfrac{2000}{100} = 20$ ⁠. And about ten such bacteria in a row span a typical human cell.

Hint
A micrometre is 1000 nm.

Worked solution
1. Put both in the same unit: 2 µm $= 2 \times 1000 = 2000$ ⁠ nanometres.
2. Divide by the length of one virus: $2000 \div 100 = 20$ ⁠.
3. The answer: 20 viruses.
If you got $0.02$ ⁠, you divided the other way round, the virus by the bacterium.
Step 3 of 7
Open this step
Taking over a cell
Practice · Predict, then look
Question
The coronavirus attaches to a cell and its genes get inside. Who will build the new copies of the virus?

Correct answer
- Correct The cell’s own machinery, following the instructions in the virus’s genes
Setup, choices and feedback
The Scene at this step
Virus
Coronavirusfixed
Infection
0 %fixed
Cut open
Whole
Choices
- Correct The cell’s own machinery, following the instructions in the virus’s genes
- Wrong The virus itself, with the enzymes it brought along
  Misconception: Thinks a virus makes copies of itself with its own machinery.
  Feedback from the catalogue
  A virus has no ribosomes, no enzymes to release energy and no raw materials. It puts its genes into a cell, and the cell’s own machinery reads them as if they were the cell’s: it copies the viral genome and builds the viral proteins. Outside a cell a virus does nothing at all.
- Wrong The virus will split in two, like a bacterium
  Misconception: Thinks a virus is just a very small bacterium.
  Feedback from the catalogue
  A bacterium is a living cell: it has a membrane, ribosomes, machinery to release energy, and it divides by itself. A virus is not a cell at all: a genome of DNA or RNA in a protein coat, with no machinery of its own. It is also about twenty times smaller than a typical bacterium. A virus can copy itself only inside someone else’s cell.
Once the prediction is locked in, the Scene changes
Infection
60 %fixed
The Scene then shows: 5 · The step of the infection cycle: 1 drifting, 2 attached, 3 getting in, 4 takeover, 5 copying, 6 assembling, 7 release/nodes/virus/outputs/phase
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
At the copying stage the cell’s ribosomes build viral proteins, and the cell copies the viral genome. The cell works for the virus until it fills up with parts, and then the new particles assemble themselves.

Worked solution
1. The virus brings only genes, and it has no ribosomes.
2. Inside the cell, the ribosomes read the viral genes as if they were the cell’s own.
3. The cell copies the genome and builds the proteins, and the parts come together as new viruses.
If you predicted the virus does this itself, you gave it machinery it does not have.
Step 4 of 7
Open this step
Genes in a box
Practice · Predict, then look
Question
The coronavirus’s genome is one long molecule of RNA. In the Influenza virus, in how many separate pieces will the genome come?

Correct answer
- Correct Eight
Setup, choices and feedback
The Scene at this step
Virus
Coronavirusfixed
Infection
0 %
Cut open
Cut open
Choices
- Wrong One, as in the coronavirus
  No feedback of its own; the learner sees the standard message
  Not yet. Look again and try once more.
- Wrong Two
  No feedback of its own; the learner sees the standard message
  Not yet. Look again and try once more.
- Correct Eight
Once the prediction is locked in, the Scene changes
Virus
Influenzafixed
Cut open
Cut open
The Scene then shows: 8 · How many separate pieces the genome comes in/nodes/virus/outputs/pieces
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Eight segments of RNA. When two strains of flu infect the same cell, their segments can mix in the new particles, which is how completely new strains sometimes arise.

Worked solution
1. Cut open at Cut open lifts the genome out of the virus.
2. In the influenza virus the genome is split into eight separate segments of RNA.
3. Each segment carries one or two genes.
It is one reason a new flu vaccine is needed almost every year.
Step 5 of 7
Open this step
Envelopes
Practice · Choose
Question
Some viruses wear an envelope: a piece of fatty membrane taken from the cell they left. Go through the viruses in the Scene and choose all that have an envelope.

Correct answer
- Correct Coronavirus
- Correct Influenza
Setup, choices and feedback
The Scene at this step
Virus
Coronavirus
Infection
0 %
Cut open
Whole
Choices
Shown in a random order. More than one choice is correct: every one must be selected.
- Correct Coronavirus
- Correct Influenza
- Wrong Adenovirus
  Feedback
  The adenovirus has only a twenty-sided protein capsid, with no envelope.
- Wrong Phage T4
  Feedback
  T4 has a protein head and tail, with no envelope: it lands on a bacterium and injects its DNA.
Feedback for a right answer
Coronavirus and flu. The fat in the envelope is their weak spot: soap breaks it apart, which is why washing hands with soap and water puts them out of action.

Hint
The envelope looks like a smooth layer studded with sticking-out proteins.

Worked solution
1. Coronavirus: a fatty envelope covered in spike proteins.
2. Influenza: an envelope with two kinds of sticking-out proteins.
3. Adenovirus and T4: a bare protein capsid, with no envelope.
The envelope is taken from the membrane of the cell the virus buds out of.
Step 6 of 7
Open this step
Why not antibiotics
Practice · Choose
Question
Flu is caused by a virus. Why do antibiotics not cure it?

Correct answer
- Correct Antibiotics hit the cell wall and ribosomes of bacteria, and a virus has neither
Setup, choices and feedback
The Scene at this step
Virus
Influenza
Infection
68 %
Cut open
Whole
Choices
Shown in a random order.
- Correct Antibiotics hit the cell wall and ribosomes of bacteria, and a virus has neither
- Wrong They do cure it, if taken early enough
  Misconception: Thinks antibiotics work against viruses, such as those that cause colds and flu.
  Feedback from the catalogue
  Antibiotics attack structures only bacteria have: their cell wall and their own kind of ribosome. A virus has no wall and no ribosomes; it uses our cells’ ribosomes. So antibiotics cannot touch viruses, and do not help with a cold or flu. Against viruses there are other drugs, antivirals, and vaccines.
- Wrong The flu virus is too strong for antibiotics
  Misconception: Thinks antibiotics work against viruses, such as those that cause colds and flu.
  Feedback from the catalogue
  Antibiotics attack structures only bacteria have: their cell wall and their own kind of ribosome. A virus has no wall and no ribosomes; it uses our cells’ ribosomes. So antibiotics cannot touch viruses, and do not help with a cold or flu. Against viruses there are other drugs, antivirals, and vaccines.
Feedback for a right answer
Right. Antibiotics harm bacteria without harming our cells, because they aim at structures only bacteria have. A virus builds none of those: it uses the machinery of our own cells.

Hint
Think about what a bacterium has and a virus lacks.

Worked solution
1. Antibiotics aim at bacterial structures: the cell wall and bacterial ribosomes.
2. A virus has no wall and no ribosomes; it builds itself with our cells’ ribosomes.
3. So antibiotics have nothing to hit.
Against viruses there are vaccines, and antiviral drugs that aim at one step in one virus’s cycle.
Step 7 of 7
Open this step
Bursting out
Practice · Calculate
Question
One T4 phage turns a bacterium into about 100 new phages in about half an hour. If each of them infects a new bacterium, how many phages will there be after two rounds?

Correct answer
10000 ± 1%
Setup, choices and feedback
The Scene at this step
Virus
Phage T4fixed
Infection
100 %
Cut open
Whole
Accepted within
± 1%
A fixed value in the question
Wrong answers it recognises
- 200 ± 1
  Misconception: Treats repeated rounds of multiplication as adding the same amount each round, instead of multiplying.
  Feedback from the catalogue
  When every copy makes several new ones, growth multiplies: each round the amount is multiplied by the same factor, not increased by the same step. One round of a hundredfold gives a hundred; two rounds give $100 \times 100 = 10\,000$ ⁠, not $100 + 100$ ⁠. This is the exponential growth of the exponents and logarithms Unit.
Feedback for a right answer
$100 \times 100 = 100^2 = 10\,000$ ⁠, in just an hour. After a third round, a million.

Hint
In each round every phage becomes a hundred.

Worked solution
1. First round: one phage becomes 100.
2. Second round: each of the 100 becomes 100: $100 \times 100 = 10\,000$ ⁠.
3. In general, after $n$ ⁠ rounds: $100^n$ ⁠.
If you answered 200, you added a hundred each round instead of multiplying by a hundred.

Lesson 2 of 4 · About 20 minutes

The immune response

A first meeting with a virus: how fast it multiplies, what antibodies are, why the virus peaks before they rise, and how many days you feel ill.

Open the lesson

The 4 steps and their answers

Step 1 of 4
Open this step
A first meeting
Practice · Calculate
Question
At a first meeting with the virus, without a vaccine, it grows from 100 copies per millilitre on day 0 to about 17,300 on day 2. By what factor does it grow each day, if the growth multiplies?

Correct answer
13.2 ± 4%
Setup, choices and feedback
The Scene at this step
Vaccine
Nonefixed
Days since infection
0.095 (Virus lands → Three weeks on)
Accepted within
± 4%
A fixed value in the question
Wrong answers it recognises
- 86.6 ± 4%
  Misconception: Treats repeated rounds of multiplication as adding the same amount each round, instead of multiplying.
  Feedback from the catalogue
  When every copy makes several new ones, growth multiplies: each round the amount is multiplied by the same factor, not increased by the same step. One round of a hundredfold gives a hundred; two rounds give $100 \times 100 = 10\,000$ ⁠, not $100 + 100$ ⁠. This is the exponential growth of the exponents and logarithms Unit.
Feedback for a right answer
$\sqrt{173} \approx 13.2$ ⁠: almost thirteenfold every day. At that rate a thousand copies become a million in under three days.

Hint
Over two days it grew $173$ ⁠-fold. What number times itself gives $173$ ⁠?

Worked solution
1. The total growth: $17\,300 \div 100 = 173$ ⁠.
2. If the virus grows by a factor $g$ ⁠ each day, over two days it grows by $g^2$ ⁠.
3. $g^2 = 173$ ⁠, so $g = \sqrt{173} \approx 13.2$ ⁠.
If you answered about $86$ ⁠, you halved the total factor, as if each day added half of it. In multiplying growth you take a root.
Step 2 of 4
Open this step
Antibodies
Practice · Choose
Question
After about a week antibodies start to appear. What is an antibody?

Correct answer
- Correct A protein made by plasma cells, which binds to one particular virus and blocks it
Setup, choices and feedback
The Scene at this step
Vaccine
None
Days since infection
0.667 (Virus lands → Three weeks on)
Choices
Shown in a random order.
- Correct A protein made by plasma cells, which binds to one particular virus and blocks it
- Wrong A white blood cell that swallows viruses
  Misconception: Thinks antibodies are cells, such as white blood cells that swallow germs.
  Feedback from the catalogue
  An antibody is a protein, not a cell: a Y-shaped molecule, thousands of times smaller than a cell. Plasma cells, which develop from B cells, make it by the thousand every second. The antibody binds to a matching virus particle, blocks it and tags it for other cells, which swallow it.
- Wrong A substance the body makes against all bacteria and viruses at once
  Feedback
  Each antibody fits one part of one particular pathogen. That is why it takes time to find and multiply the cells that make the right one.
Feedback for a right answer
Right. Out of millions of B cells, only the few whose antibody fits the virus divide and become plasma cells. That takes about a week, which is why the first meeting is slow.

Hint
The small Y-shaped specks in the Scene are antibodies.

Worked solution
1. Each B cell carries one shape of antibody on its surface.
2. A B cell whose antibody fits the virus divides again and again, and some of its offspring become plasma cells.
3. Plasma cells release the antibody, a protein, into the blood and the lining of the airways.
If you chose “a white blood cell”, you mixed up the antibody with the cells that make it or that swallow what it tags.
Step 3 of 4
Open this step
The race
Practice · Predict, then look
Question
At a first meeting, without a vaccine, the virus multiplies and antibodies start to appear. Which will come first?

Correct answer
- Correct The virus will peak, and only then will the antibodies climb
Setup, choices and feedback
The Scene at this step
Vaccine
Nonefixed
Days since infection
Virus lands
Choices
- Correct The virus will peak, and only then will the antibodies climb
- Wrong Antibodies will rise at once and stop the virus before it multiplies
  Misconception: Thinks the immune system answers a germ the same way each time it meets it.
  Feedback from the catalogue
  After a first meeting the body keeps memory cells: B and T cells that already know the virus. At the second meeting they divide and make antibodies at once, without a week of searching, so the response is much faster and stronger. Usually the virus is cleared before you feel ill.
Once the prediction is locked in, the Scene changes
Days since infection
Three weeks on
The Scene then shows: 6.655 · The day the virus peaks/nodes/immune/outputs/peakDay
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The virus peaks around day 6.7, while the antibodies are still almost zero. They rise only at the end of the first week, and by day 14 they are hundreds of times higher. At a first meeting the body lags behind the virus, and that is when you feel ill.

Worked solution
1. In the first days the virus multiplies thirteenfold a day, and the immune system is still searching for the matching cells.
2. The virus peaks on day 6–7.
3. Only then do the antibodies start to rise, and the virus is cleared by day 9–10.
At a second meeting the order flips: memory answers before the virus has time to multiply.
Step 4 of 4
Open this step
Days ill
Practice · Calculate
Question
Without a vaccine, how many days in all do you feel ill, with the virus above the level that makes you feel unwell? Read it at the foot of the chart, on day 21.

Correct answer
5.93 d ± 0.3 d
Setup, choices and feedback
The Scene at this step
Vaccine
Nonefixed
Days since infection
Three weeks onfixed
Accepted within
± 0.3 d
Computed from the Scene at this step’s setting
Time spent ill over the three weeks/nodes/immune/outputs/illDuration
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
About 6 days, from roughly day 3.5 to day 9.5. That is how long the body takes to find the matching cells, multiply them and clear the virus.

Hint
In the upper chart, the dashed line is the level above which you feel ill.

Worked solution
1. The virus crosses the dashed line around day 3.5.
2. It drops below it as the antibodies take over, around day 9.5.
3. The difference: about 6 days.
The foot of the chart says how many days the body has been ill up to the day shown.

Lesson 3 of 4 · About 25 minutes

Vaccines and immune memory

A second meeting with the same virus: memory cells, a vaccine from last month and from ten years ago, how low the virus stays, and how many antibodies a secondary response makes.

Open the lesson

The 6 steps and their answers

Step 1 of 6
Open this step
A second meeting
Practice · Predict, then look
Question
Without a vaccine you feel ill for about six days. What will happen if the same person had the vaccine Last month?

Correct answer
- Correct They will not feel ill at all
Setup, choices and feedback
The Scene at this step
Vaccine
Nonefixed
Days since infection
Three weeks on
Choices
- Correct They will not feel ill at all
- Wrong They will be ill, but a day or two less
  No feedback of its own; the learner sees the standard message
  Not yet. Look again and try once more.
- Wrong The same: the immune system will need a week again
  Misconception: Thinks the immune system answers a germ the same way each time it meets it.
  Feedback from the catalogue
  After a first meeting the body keeps memory cells: B and T cells that already know the virus. At the second meeting they divide and make antibodies at once, without a week of searching, so the response is much faster and stronger. Usually the virus is cleared before you feel ill.
Once the prediction is locked in, the Scene changes
Vaccine
Last monthfixed
Days since infection
Three weeks on
The Scene then shows: 0 d · Time spent ill over the three weeks/nodes/immune/outputs/illDuration
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Zero days ill. The virus reaches only about 270 copies and is gone by day 3, because antibodies are already waiting and the memory cells boost them at once.

Worked solution
1. The vaccine taught the immune system the virus without the illness, and left antibodies and memory cells.
2. When the virus arrives, the antibodies in the blood block some of it at once, and the memory cells make more within a few days.
3. The virus never reaches the level that makes you feel ill.
If you predicted “the same”, you left out immune memory.
Step 2 of 6
Open this step
Ten years on
Practice · Predict, then look
Question
A vaccine Ten years ago: on the day the virus lands there are only 2 units of antibody in the blood, against 30 after last month’s vaccine. Will the person fall ill?

Correct answer
- Correct No: the memory cells will answer fast enough
Setup, choices and feedback
The Scene at this step
Vaccine
Ten years agofixed
Days since infection
Virus lands
Choices
- Correct No: the memory cells will answer fast enough
- Wrong Yes, like someone never vaccinated, because the antibodies are almost gone
  Misconception: Thinks a vaccine stops protecting once the antibodies it made have faded from the blood.
  Feedback from the catalogue
  The antibodies a vaccine makes do fade over the months and years after it. But the memory cells stay, sometimes for decades. When the virus arrives, they make new antibodies within a few days, fast enough to stop it before it multiplies much. The protection is in the memory, not only in the antibodies in the blood.
Once the prediction is locked in, the Scene changes
Days since infection
Three weeks on
The Scene then shows: 0 d · Time spent ill over the three weeks/nodes/immune/outputs/illDuration
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Zero days ill. The virus reaches about 52,500 copies, more than after a fresh vaccine, but the memory cells make antibodies within a few days and clear it by day 4.5, before it crosses the level of illness.

Worked solution
1. The antibodies the vaccine made have faded over ten years.
2. The memory cells stayed. When the virus arrives, they divide and make new antibodies within days, not a week.
3. The virus is cleared before it reaches the level that makes you feel ill.
If you predicted illness, you thought the protection was only the antibodies in the blood.
Step 3 of 6
Open this step
How high it gets
Practice · Calculate
Question
Without a vaccine the virus peaks at about 3,750,000 copies per millilitre; with a vaccine ten years ago, at about 52,500. How many times lower is the peak with the old vaccine?

Correct answer
71.5 ± 4%
Setup, choices and feedback
The Scene at this step
Vaccine
Ten years agofixed
Days since infection
Three weeks on
Accepted within
± 4%
A fixed value in the question
Feedback for a right answer
$\dfrac{3\,750\,000}{52\,500} \approx 71$ ⁠. At thirteenfold a day, that is like stopping the virus almost two days sooner.

Hint
Divide the higher peak by the lower one.

Worked solution
1. The peak without a vaccine: 3,750,000.
2. The peak with the old vaccine: 52,500.
3. The ratio: $3\,750\,000 \div 52\,500 \approx 71$ ⁠.
The chart shows the virus on a logarithmic scale, so the gap looks like only about two gridlines: each line is a factor of ten.
Step 4 of 6
Open this step
A stronger answer
Practice · Predict, then look
Question
At a first meeting the antibodies reach about 80 units on day 21. How many units will there be on day 21 after the virus meets someone vaccinated ten years ago? Predict a number.

Correct answer
860.1 ± 50%
A number prediction counts as right within ± 50%
Setup, choices and feedback
The Scene at this step
Vaccine
Ten years agofixed
Days since infection
Virus lands
Once the prediction is locked in, the Scene changes
Days since infection
Three weeks on
The Scene then shows: 860.1 · Antibodies on the day shown, relative units (a first infection peaks near 80)/nodes/immune/outputs/antibodies
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
About 860 units, roughly ten times the first meeting. Each meeting with the virus, or a booster dose, enlarges the pool of memory and improves the antibodies.

Worked solution
1. At a first meeting the body starts from a few matching cells, and reaches about 80 units.
2. At a second meeting there are many more memory cells, and their antibodies are already improved.
3. They make antibodies fast and in bulk: about 860 units on day 21.
This is the secondary response: faster and larger than the first.
Step 5 of 6
Open this step
Staying well
Practice · Reach the target
Question
Choose a Vaccine setting in which the person does not feel ill for a single day, and press Check.

Goal
- 0 d ± 0.01 d
  At the start of the step: 5.93 d
Setup, choices and feedback
The Scene at this step
Vaccine
None
Days since infection
Three weeks on
Read from the Scene at
Time spent ill over the three weeks/nodes/immune/outputs/illDuration
One setting that reaches it
Vaccine
Last month
Feedback for a right answer
Both last month’s vaccine and one from ten years ago prevent every day of illness in this model. The difference between them is how high the virus climbs before it is cleared.

Feedback for a wrong answer
Not yet. Without a vaccine you feel ill for about six days.

Hint
The foot of the chart says how many days the body was ill.

Worked solution
1. None: about six days ill.
2. Last month: zero, the virus reaches only about 270 copies.
3. Ten years ago: zero too, although the virus reaches about 52,500.
Both vaccinated settings meet the goal, because memory cells clear the virus before the level of illness.
Step 6 of 6
Open this step
What a vaccine does
Practice · Choose
Question
How does a vaccine prepare the body for a virus?

Correct answer
- Correct It shows the immune system something that looks like the virus but does not cause the disease, and it builds memory cells
Setup, choices and feedback
The Scene at this step
Vaccine
Last month
Days since infection
Three weeks on
Choices
Shown in a random order.
- Correct It shows the immune system something that looks like the virus but does not cause the disease, and it builds memory cells
- Wrong It gives a mild case of the disease
  Misconception: Thinks a vaccine works by giving a mild case of the disease.
  Feedback from the catalogue
  A vaccine shows the immune system something that looks like the virus without causing the disease: a killed virus, a weakened one, one of its proteins, or the building plan for that protein. The immune system learns to recognise it and builds memory cells, without going through the illness.
- Wrong It is a drug that kills the virus when it arrives
  Feedback
  A vaccine does not act on the virus itself; it teaches the immune system. The protection is the body’s own, which is why it lasts years after the vaccine itself is gone.
Feedback for a right answer
Right. The vaccine gives the body its first, slow meeting, without the danger. When the real virus arrives, it is already a second meeting.

Hint
Think about what stays in the body after a first meeting.

Worked solution
1. A vaccine holds a killed or weakened virus, one protein of the virus, or the building plan for it.
2. The immune system learns to recognise it and builds antibodies and memory cells.
3. When the real virus arrives, the answer is a secondary response: fast and strong.
If you chose “a mild case of the disease”, you thought learning needs illness. The vaccine gives the learning without the illness.

Lesson 4 of 4 · About 30 minutes

Epidemics and the SIR model

How a disease spreads through a crowd: exponential growth over generations of infection, the basic and effective reproduction numbers, the herd-immunity threshold, measles, and why vaccination also protects the unvaccinated.

Open the lesson

The 7 steps and their answers

Step 1 of 7
Open this step
Chains of infection
Practice · Calculate
Question
Each case infects $R_0 = 2.5$ ⁠ people on average. Start from one case. How many new cases will there be in the fourth generation of infection, if everyone is still susceptible?

Correct answer
39.06 ± 1
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
2.5
Vaccinated
0 %
Days
0.25 (First case → Day 120)
Accepted within
± 1
A fixed value in the question
Wrong answers it recognises
- 10 ± 0.5
  Misconception: Treats repeated rounds of multiplication as adding the same amount each round, instead of multiplying.
  Feedback from the catalogue
  When every copy makes several new ones, growth multiplies: each round the amount is multiplied by the same factor, not increased by the same step. One round of a hundredfold gives a hundred; two rounds give $100 \times 100 = 10\,000$ ⁠, not $100 + 100$ ⁠. This is the exponential growth of the exponents and logarithms Unit.
- 11 ± 0.5
  Misconception: Treats repeated rounds of multiplication as adding the same amount each round, instead of multiplying.
  Feedback from the catalogue
  When every copy makes several new ones, growth multiplies: each round the amount is multiplied by the same factor, not increased by the same step. One round of a hundredfold gives a hundred; two rounds give $100 \times 100 = 10\,000$ ⁠, not $100 + 100$ ⁠. This is the exponential growth of the exponents and logarithms Unit.
Feedback for a right answer
$2.5^4 \approx 39$ ⁠. Four generations more, and it is already about 1500: this is the exponential growth at the start of an epidemic.

Hint
Each generation the number of new cases is multiplied by $2.5$ ⁠.

Worked solution
1. Generation 1: $2.5$ ⁠ cases.
2. Each generation multiplies by $2.5$ ⁠: generation 2 is $2.5^2 = 6.25$ ⁠, generation 3 is $2.5^3 \approx 15.6$ ⁠.
3. Generation 4: $2.5^4 \approx 39$ ⁠.
If you answered $10$ ⁠, you added $2.5$ ⁠ each generation instead of multiplying.
Step 2 of 7
Open this step
Faster spread
Practice · Predict, then look
Question
With $R_0 = 2.5$ ⁠ and no vaccine, the number ill at once peaks around day 20. What will happen to the peak when R₀: people one case infects rises to 4?

Correct answer
- Correct It will come sooner and be higher
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
2.5
Vaccinated
0 %fixed
Days
Day 120
Choices
- Correct It will come sooner and be higher
- Wrong It will come on the same day, only higher
  No feedback of its own; the learner sees the standard message
  Not yet. Look again and try once more.
- Wrong It will come later, because more people will fall ill
  No feedback of its own; the learner sees the standard message
  Not yet. Look again and try once more.
Once the prediction is locked in, the Scene changes
R₀: people one case infects
4
The Scene then shows: 14.75 · The day the most people are ill at once/nodes/crowd/outputs/peakDay
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The peak comes around day 15, with about 130 of 300 ill at once, against 76. Faster growth reaches the point where there are no longer enough susceptible people sooner.

Worked solution
1. With a larger $R_0$ ⁠ each generation multiplies by more, and the numbers grow faster.
2. The peak comes when only $\dfrac{1}{R_0}$ ⁠ of the crowd is still susceptible: each case then infects one on average.
3. With $R_0 = 4$ ⁠ that point comes sooner, after more people have already fallen ill.
On the chart, the dotted line at $\dfrac{1}{R_0}$ ⁠ marks that point.
Step 3 of 7
Open this step
The effective number
Practice · Calculate
Question
$R_0 = 2.5$ ⁠, and 30% of the crowd are Vaccinated. How many people does each case infect on average at the start of the outbreak?

Correct answer
1.75 ± 0.02
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
2.5fixed
Vaccinated
30 %fixed
Days
Day 120
Accepted within
± 0.02
Computed from the Scene at this step’s setting
R at the start: R₀ times the share not vaccinated/nodes/crowd/outputs/reff
Wrong answers it recognises
- 0.75 ± 0.02
  Misconception: Multiplies R₀ by the share vaccinated, or subtracts the share from R₀, instead of multiplying by the share still susceptible.
  Feedback from the catalogue
  A case infects only people who can still catch it. If a share $v$ ⁠ is vaccinated, a share $1 - v$ ⁠ is left susceptible, so $R = R_0 \times (1 - v)$ ⁠. With $R_0 = 2.5$ ⁠ and 30% vaccinated: $2.5 \times 0.7 = 1.75$ ⁠. Not $2.5 \times 0.3$ ⁠, which is the part the vaccine stopped, and not $2.5 - 0.3$ ⁠.
- 2.2 ± 0.02
  Misconception: Multiplies R₀ by the share vaccinated, or subtracts the share from R₀, instead of multiplying by the share still susceptible.
  Feedback from the catalogue
  A case infects only people who can still catch it. If a share $v$ ⁠ is vaccinated, a share $1 - v$ ⁠ is left susceptible, so $R = R_0 \times (1 - v)$ ⁠. With $R_0 = 2.5$ ⁠ and 30% vaccinated: $2.5 \times 0.7 = 1.75$ ⁠. Not $2.5 \times 0.3$ ⁠, which is the part the vaccine stopped, and not $2.5 - 0.3$ ⁠.
Feedback for a right answer
$2.5 \times 0.7 = 1.75$ ⁠. Still above one, so the outbreak grows, but more slowly: the peak is lower and later.

Hint
A case infects only those who can still catch it. What share of the crowd is susceptible?

Worked solution
1. Vaccinated: 30%, so susceptible: $1 - 0.3 = 0.7$ ⁠.
2. The effective reproduction number: $R = R_0 \times (1 - v) = 2.5 \times 0.7$ ⁠.
3. $R = 1.75$ ⁠.
If you answered $0.75$ ⁠, you multiplied by the vaccinated share. If you answered $2.2$ ⁠, you subtracted a percentage from a reproduction number.
Step 4 of 7
Open this step
The threshold
Practice · Calculate
Question
For a disease with $R_0 = 4$ ⁠: what percentage of the crowd must be immune so that each case infects one person or fewer on average?

Correct answer
75 ± 0.5
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
4fixed
Vaccinated
0 %fixed
Days
Day 120
Accepted within
± 0.5
Computed from the Scene at this step’s setting
Per cent who must be immune for cases to fall, 100 (1 − 1/R₀)/nodes/crowd/outputs/herdThreshold
Wrong answers it recognises
- 25 ± 0.5
  Misconception: Works out the herd-immunity threshold as 1/R₀ instead of 1 − 1/R₀.
  Feedback from the catalogue
  $\dfrac{1}{R_0}$ ⁠ is the share that may stay susceptible: if each case meets $R_0$ ⁠ people and a share $\dfrac{1}{R_0}$ ⁠ of them can catch it, they infect exactly one on average. So the share that must be immune is the rest, $1 - \dfrac{1}{R_0}$ ⁠. With $R_0 = 4$ ⁠ a quarter may stay susceptible, and three-quarters must be immune.
- 100 ± 0.5
  Misconception: Thinks an outbreak can be stopped only if everyone is vaccinated.
  Feedback from the catalogue
  Not everyone is needed. An outbreak dies away when each case infects fewer than one person on average, and that happens once a share $1 - \dfrac{1}{R_0}$ ⁠ of the population is immune. For seasonal flu that is under a third, for COVID in 2020 about sixty per cent, and for measles, one of the most contagious diseases, about 93%.
Feedback for a right answer
$1 - \dfrac{1}{4} = 0.75$ ⁠, that is 75%. This is the herd-immunity threshold: above it every chain of infection dies away.

Hint
Solve $4 \times (1 - v) = 1$ ⁠.

Worked solution
1. You want $R = R_0 \times (1 - v) \le 1$ ⁠.
2. $1 - v \le \dfrac{1}{R_0} = \dfrac{1}{4}$ ⁠.
3. $v \ge 1 - \dfrac{1}{4} = 0.75$ ⁠, that is 75%.
If you answered 25%, you gave the share that may stay susceptible. If you answered 100%, you thought everyone has to be vaccinated.
Step 5 of 7
Open this step
Stopping an outbreak
Practice · Reach the target
Question
R₀: people one case infects is fixed at 2.5. Change Vaccinated until each case infects fewer than one person on average, but no more than needed, and press Check.

Goal
- 0.95 ± 0.05
  At the start of the step: 2.5
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
2.5fixed
Vaccinated
0 %
Days
Day 120
Read from the Scene at
R at the start: R₀ times the share not vaccinated/nodes/crowd/outputs/reff
One setting that reaches it
Vaccinated
60 %
Feedback for a right answer
From 60% up: $1 - \dfrac{1}{2.5} = 0.6$ ⁠. Move Days forward: the first case infects one or two, and the chain breaks.

Feedback for a wrong answer
Not yet. The reproduction number has to fall to 1 or just below, but not far below.

Hint
$1 - \dfrac{1}{R_0}$ ⁠.

Worked solution
1. $R = 2.5 \times (1 - v)$ ⁠.
2. For $R \le 1$ ⁠: $1 - v \le 0.4$ ⁠, so $v \ge 0.6$ ⁠.
3. Set Vaccinated to 60% or a little more.
That is the herd-immunity threshold for COVID in 2020: about sixty per cent.
Step 6 of 7
Open this step
Measles
Practice · Reach the target
Question
Measles is one of the most contagious diseases: R₀: people one case infects is fixed at 15. Find the smallest whole percentage Vaccinated at which each case infects fewer than one person, and press Check.

Goal
- 0.95 ± 0.06
  At the start of the step: 7.5
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
15fixed
Vaccinated
50 %
Days
Day 120
Read from the Scene at
R at the start: R₀ times the share not vaccinated/nodes/crowd/outputs/reff
One setting that reaches it
Vaccinated
94 %
Feedback for a right answer
94%: $1 - \dfrac{1}{15} \approx 0.933$ ⁠, and in whole percentages 94%. That is why the vaccination target for measles is so high, and a small fall in coverage is enough for measles to return.

Feedback for a wrong answer
Not yet. Work out $1 - \dfrac{1}{15}$ ⁠ and round up.

Hint
$1 - \dfrac{1}{15}$ ⁠.

Worked solution
1. $v \ge 1 - \dfrac{1}{15} \approx 0.933$ ⁠.
2. At 93%: $R = 15 \times 0.07 = 1.05$ ⁠, still above one.
3. At 94%: $R = 15 \times 0.06 = 0.9$ ⁠, below one.
When each case infects fifteen, nearly everyone has to be protected, but still not everyone.
Step 7 of 7
Open this step
Herd immunity
Practice · Predict, then look
Question
$R_0 = 2.5$ ⁠, and 70% of the crowd are vaccinated. About 90 of the 300 people are not. How many people will have caught it by day 120?

Correct answer
- Correct A handful, fewer than ten
Setup, choices and feedback
The Scene at this step
R₀: people one case infects
2.5fixed
Vaccinated
70 %fixed
Days
First case
Choices
- Correct A handful, fewer than ten
- Wrong Most of the unvaccinated, about 70 to 90
  Misconception: Thinks vaccination protects only the people who are vaccinated.
  Feedback from the catalogue
  When enough people are immune, the virus cannot find enough susceptible people to keep spreading, and the chains break. So even people who are not vaccinated, such as babies too young for the vaccine or people with weak immune systems, are hardly ever infected. This is herd immunity.
Once the prediction is locked in, the Scene changes
Days
Day 120
The Scene then shows: 6 · People who have caught it by the day shown/nodes/crowd/outputs/ever
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Only 6 people. Each case infects $2.5 \times 0.3 = 0.75$ ⁠ on average, so every chain dies out quickly, and the unvaccinated are protected by the vaccinated around them.

Worked solution
1. $R = 2.5 \times (1 - 0.7) = 0.75$ ⁠, less than one.
2. The first case infects fewer than one person on average, who infects fewer still.
3. The chain stops after a few cases, all among the unvaccinated, and most of the unvaccinated never meet it.
If you predicted that most of the unvaccinated would fall ill, you thought a vaccine protects only those who get it. Above the threshold it protects the rest too.

Across the unit · 12 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

Viruses

Not a cell

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

How small

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Taking over a cell

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Genes in a box

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Envelopes

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Why not antibiotics

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Bursting out

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

The immune response

A first meeting

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Antibodies

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

The race

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Days ill

Question

Correct answer

The Scene at this step

Hint

Worked solution

Vaccines and immune memory

A second meeting