Life saving maths: How does vaccination work?

In 1998, Andrew Wakefield published an article in The Lancet, claiming that he had discovered a link between the MMR vaccine (for measles, mumps and rubella) and autism.  Many parents decided not to have their children vaccinated, despite the scientific consensus that there is no evidence of such a link.

As a result the number of cases of measles and mumps rose sharply on rates prior to 2000.

It has never been necessary for 100% of children to be vaccinated, and there are some for whom it is definitely not in their best interests, such as children with any disease that affects their immune system, and those who have received a transplant organ.  Provided a large enough proportion of children are vaccinated, however, everyone is protected - including those who can't be vaccinated.  This is known as herd immunity.

But how does this work?  How do we know what the rate should be?  For measles, it is 95% - how do doctors know that?

Mathematical modelling provides the answer, helping us to understand how epidemics spread and how vaccination prevents the spread of disease.  In the video clips in this pack, Dr Julia Gog and Dr Andrew Conlan, of the University of Cambridge, explain using simple models.  The accompanying activities will help students to understand how these models work, and how mathematicians can help policy makers to make sensible decisions.

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The resources in this pack complement the video clips, providing activities designed to help students understand how the maths they are taught in school can help them to become better informed about such issues.  Answers and additional notes are also provided.

The video clips and easier follow-up resources labelled * can be used with most 13-15 year-old secondary students.  Those labelled *** may be more suitable for older students preparing for public exams, such as GCSE or equivalent, at higher levels.

Any of these resources can be used alone - although students will find it easier to understand them if they have seen the preceding video clip(s). 

• Download all videos (zip file, 92MB)
• Download all pdf files

Type of Resource	Resource Name	Difficulty	Notes
	Teacher notes and worksheet answers		Start here!
	How does vaccination work: Introduction	*	Introductory video clip (3 min 38 secs) - Dr Andrew Conlan
	The measles graph	*	Follow-up discussion on patterns in the measles graph
	Measles and vaccination	**	Video clip (5 mins 53 secs) - Dr Andrew Conlan
	Standing Disease	*	Activity: a simple model of epidemic spread
	Graphing Change	**	Worksheet: three contexts (including Standing Disease) comparing different scale representations of the data, and three different representations of the measles data discussed in the first two video clips.
	Counter Plague	*	Activity: varying the effect of a contact
	Counter Plague: demonstration	*	Video clip (3 min 47 secs) - you could omit this if the activity instructions are clear without it, or use it as a teacher resource.
	26-Card Disease	**	Activity: Refining the model to take account of immunity (natural or induced through vaccination)
	26-Card Disease: demonstration	**	Video clip (7 mins 37 secs) - you could omit this if the activity instructions are clear without it, or use it as a teacher resource.
	Reproductive Ratio	**	Video clip (3 mins 56 secs) - Dr Andrew Conlan
	Calculating R0: Counter Plague	**	Presentation: how do we calculate the reproductive ratio for Counter Plague?
	Immunisation and Reproductive Ratio	**	Video clip (3 mins 41 secs) - Dr Julia Gog
	How many people do we need to vaccinate?	**	Presentation: how do we calculate the proportion of the population which needs to be vaccinated to produce herd immunity?
	e-Counter Plague	**	Activity: a set of 4 online simulations to help students investigate the effect of vaccination.
	Who should be immunised?	*	Video clip (2 mins 10 secs) - Dr Julia Gog
	Network Disease	*	Activity: identifying the right people to vaccinate.
	Investigating Networks	***	Worksheet: identifying the key person in a network.