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# Vaccination Game

This activity invites students to consider vaccination strategies, and how many people should be vaccinated in a population, based on the $R_0$ of a pathogen, in order to limit the impact of an outbreak.

You do not need to vaccinate a whole population in order to control a (future) outbreak of disease, but that if a certain proportion of the population has been vaccinated, this will offer herd immunity to the remaining population. In a large population, with people immune to catching the disease (due to vaccination), the chains of transmission will be broken, and others susceptible individuals in the community will not become infected.

The proportion of the population that needs to be vaccinated, also known as the vaccine threshold, can be calculated from the $R_0$ of the pathogen:$$\frac{R_0-1}{R_0}$$As such, for a disease with $R_0 = 2$ (flu, ebola etc), you need to vaccinate at least $50\%$ of the population. But when $R_0 = 18$ (measles), it requires $94\%$ (or $\frac{17}{18}$) of the population to be vaccinate in order to stop outbreaks.

**Resources:** Slides (Powerpoint or PDF)

One chess board per group

Three sets of counters in different colours per group (Counted in advance; distributed as 32/32, 48/16 and 56/8)

**Curriculum Link**

Maths:

### Aims

### Activity (Small Groups)

**Preparation**: Count out counters into small bags of 32/32, 48/16, 56/8 colour split between susceptible and vaccinated, and use a draughts/chess board with 64 squares for each group.

Divide the class into small groups and give each group a chess board and a number of (pre-counted) counters.

Start with 32/32 split of colours (representing 50% vaccine threshold).

Get the students to randomly spread the counters on the board, so that each square has a counter on it - and the colours are mixed.

Pick a starting point and remove that counter to represent first case. Infect susceptible neighbours (vertically and horizontally but not diagonally) and remove them from the board (ie. $R_0 =4$).

Repeat until the outbreak is over.

Count the total number of cases in the outbreak.

Repeat exercise to show that number of cases can change based on starting point of outbreak - and distribution of 'vaccinated' counters.

Repeat the exercise with 48/16 (75% threshold) and 56/8 (87.5% vaccine threshold) splits. Count the total numbers of cases in each outbreak depending on threshold.

These can then be plotted onto a graph to show frequency of each size outbreak (for the whole class) or individually to show the numbers of those infected in each simulation.

### Questions for thought

**Why did some susceptible counters not get infected during the simulations of the outbreak?**

*Herd immunity - not everyone in the community needs to get vaccinated to benefit from it.*

**Why did the size of the outbreak vary depending on where you started and how the counters were distributed?**

**What would happen if you could also infect the counters at diagonal angles from the infected case?**

*This would increase the number of susceptible people in the population, and would increase the $R_0$ to $8$. As such, you would need to have a higher vaccination threshold to reach herd immunity.*

**What would happen if we had a higher vaccine threshold?**

**Is 75% vaccination threshold enough to make this outbreak die out?**

$R_0 = 4$ as each counter can only infect those horizontally/vertically from them, so it should be!

**What are the problems with basing the health of a population on herd immunity?**

*What if everyone assumes that the rest of the population is vaccinated and doesn't do it - the free rider problem?*

**What if people don't want to get vaccinated? How does this affect vaccination policy?**

Follow-Up Activity (Whole Class)

Discuss with class how many people they think should be vaccinated to stop an outbreak in such an outbreak (with $R_0 = 4$).

Remind students of the concept of $R_0$ (from The $R_0$ Game).

Ask students how the value of $R_0$ might affect how many people should be vaccinated to stop an outbreak

Work with students to reach herd immunity (the proportion of population we need to vaccinate to stop outbreak). This is:

Refer back to the list of diseases and their R0 from previous exercise and ask students to calculate what proportion of the population they should vaccinate to get suitable herd immunity.(Remove Rabies as $R_0 = 0$.)

### Measles re-emerges in South Wales

With an $R_0$ of $15-18$, the vaccine threshold needed to offer herd immunity to the whole population of the UK is up to $94\%$ of people vaccinated.

Since 1988, UK governmental policy has been to vaccinate babies as part of their routine immunisations, and then again when they start school. However, controversy linking the MMR vaccine to an increased risk of autism and bowel problems caused several parents to choose to not have their child vaccinated, prompting fears of a future outbreak.

Public health practitioners feared that with declining rates of vaccination, the threshold required to reach herd immunity would not be met, leaving a vulnerable, susceptible population.

These fears were realised in South Wales in 2013 when an outbreak of measles spread rapidly and over 1000 children became infected with measles. This was linked to the local media at the time campaigning against the vaccine meaning that it is estimated that only $78\%$ of the population were vaccinated, which does not meet the required threshold to offer herd immunity to the community.

The study linking the vaccine to increased risk of autism and other health concerns has since been widely discredited. Furthermore, the author of this research was eventually struck of the General Medical Council Register for bringing the profession into disrepute.

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Age 11 to 16

- Project

*This classroom activity is part of the Disease Dynamics collection*

This activity invites students to consider vaccination strategies, and how many people should be vaccinated in a population, based on the $R_0$ of a pathogen, in order to limit the impact of an outbreak.

You do not need to vaccinate a whole population in order to control a (future) outbreak of disease, but that if a certain proportion of the population has been vaccinated, this will offer herd immunity to the remaining population. In a large population, with people immune to catching the disease (due to vaccination), the chains of transmission will be broken, and others susceptible individuals in the community will not become infected.

The proportion of the population that needs to be vaccinated, also known as the vaccine threshold, can be calculated from the $R_0$ of the pathogen:$$\frac{R_0-1}{R_0}$$As such, for a disease with $R_0 = 2$ (flu, ebola etc), you need to vaccinate at least $50\%$ of the population. But when $R_0 = 18$ (measles), it requires $94\%$ (or $\frac{17}{18}$) of the population to be vaccinate in order to stop outbreaks.

One chess board per group

Three sets of counters in different colours per group (Counted in advance; distributed as 32/32, 48/16 and 56/8)

Maths:

- Substituting numerical values into formulae and expressions
- Modelling situations or procedures
- Apply the concepts of instantaneous and average rate of change

- To understand how vaccination can stop a disease spreading through an otherwise susceptible population.
- To understand how a change in vaccine threshold alters how a disease spreads through the population.
- To understand how herd immunity works.
- To estimate vaccine thresholds needed for diseases based on their $R_0$ to control a disease

Divide the class into small groups and give each group a chess board and a number of (pre-counted) counters.

Start with 32/32 split of colours (representing 50% vaccine threshold).

Get the students to randomly spread the counters on the board, so that each square has a counter on it - and the colours are mixed.

Pick a starting point and remove that counter to represent first case. Infect susceptible neighbours (vertically and horizontally but not diagonally) and remove them from the board (ie. $R_0 =4$).

Repeat until the outbreak is over.

Count the total number of cases in the outbreak.

Repeat exercise to show that number of cases can change based on starting point of outbreak - and distribution of 'vaccinated' counters.

Repeat the exercise with 48/16 (75% threshold) and 56/8 (87.5% vaccine threshold) splits. Count the total numbers of cases in each outbreak depending on threshold.

These can then be plotted onto a graph to show frequency of each size outbreak (for the whole class) or individually to show the numbers of those infected in each simulation.

$R_0 = 4$ as each counter can only infect those horizontally/vertically from them, so it should be!

Follow-Up Activity (Whole Class)

Discuss with class how many people they think should be vaccinated to stop an outbreak in such an outbreak (with $R_0 = 4$).

Remind students of the concept of $R_0$ (from The $R_0$ Game).

Ask students how the value of $R_0$ might affect how many people should be vaccinated to stop an outbreak

Work with students to reach herd immunity (the proportion of population we need to vaccinate to stop outbreak). This is:

$\frac{R_0-1}{R_0}$

Refer back to the list of diseases and their R0 from previous exercise and ask students to calculate what proportion of the population they should vaccinate to get suitable herd immunity.(Remove Rabies as $R_0 = 0$.)

With an $R_0$ of $15-18$, the vaccine threshold needed to offer herd immunity to the whole population of the UK is up to $94\%$ of people vaccinated.

Since 1988, UK governmental policy has been to vaccinate babies as part of their routine immunisations, and then again when they start school. However, controversy linking the MMR vaccine to an increased risk of autism and bowel problems caused several parents to choose to not have their child vaccinated, prompting fears of a future outbreak.

Public health practitioners feared that with declining rates of vaccination, the threshold required to reach herd immunity would not be met, leaving a vulnerable, susceptible population.

These fears were realised in South Wales in 2013 when an outbreak of measles spread rapidly and over 1000 children became infected with measles. This was linked to the local media at the time campaigning against the vaccine meaning that it is estimated that only $78\%$ of the population were vaccinated, which does not meet the required threshold to offer herd immunity to the community.

The study linking the vaccine to increased risk of autism and other health concerns has since been widely discredited. Furthermore, the author of this research was eventually struck of the General Medical Council Register for bringing the profession into disrepute.