Swine FluSwine Flu

How Diseases SpreadHow Diseases Spread

Imagine you told your secret the two of your best friends and your friends passed on your secret to two of their best friends and this happened every hour

How many people would know your secret after 24 hours?1

321 7421

158421

123 2 127 3

1215 4

1224 215,777,16

That’s about a quarter of the population of the UK who know your secret!!!!

Draw a graph of ‘Number of Hours’ (plot ‘n’ on the x-axis) versus ‘Total Number of People who Know your Secret’ (plot ‘P’ on the y-axis)

Take care when scaling your axes – make sure you know what numbers you need to plot

Fully label your graph (axes, titles, etc)

What does the graph look like – what shape is it?

12 nP

Imagine you were slightly less discrete and told your secret the three of your best friends who then passed on your secret to three of their best friends and this happened every hour

How many people would know your secret after 24 hours?1

431 13931

4027931 2

13

n

Total

2

1324 )3(10141 9 SF

141 BILLION!!! That’s more than the population of the world (approx 7 billion) – AND YOU ONLY TOLD 3 PEOPLE!!!!!

How would the graph of ‘Number of Hours’ (x-axis) versus ‘Total Number of People who Know your Secret’ (y-axis) look different from the one you drew earlier?

Draw this graph

Take care when scaling your axes – make sure you know what numbers you need to plot

Fully label your graph (axes, titles, etc)2

13

n

P

What formula would you use to calculate the total number of people knowing your secret if you tell 4 people who then each go on to tell another 4 people etc?

What about 5 people, 6 people etc?

3

14

n

P

4

15

n

P5

16

n

P

You have now worked out that the total number of people who get to know your secret (we have been calling this ‘P’) after a number of hours (we have been calling this ‘n’) when each person tells your secret to 2 people, can be calculated using the formula:

12 nPWhat formula would you use to calculate the total number of people who get to know your secret if each person tells 3 people instead of 2?

What about 4 people?

What about 5 people?

What about ‘x’ people?

2

13

n

P

3

14

n

P

4

15

n

P

1

1

x

xP

n

Hours

(n)

New Peop

le

Total(P)

1 1 1

2 2 3

3 4 7

4 8 15

5 16 31

6 32 63

2 People

Hours

(n)

New Peop

le

Total(P)

1

2

3

4

5

6

3 People

Hours

(n)

New Peop

le

Total(P)

1

2

3

4

5

6

4 People

12 nP P P

Clearly this model is not perfect for our gossip spreading investigation but it does serve as a good demonstration of how numbers can theoretically grow very big very quickly

The reality is that people get bored with the secret or they are not interested or they cannot find someone who doesn’t already know the secret so they either don’t pass on your secret or they tell fewer than 3 people

The spread of your secret then starts to slow down

For the purposes of our investigation we will refer to the measure of the ability of a rumour to spread as the SPREAD FACTOR

Imagine that so far 64 people know your secret, by now it’s no longer a secret, people are becoming bored and there are fewer people to tell because many people already know the secret

You are going to investigate what happens when some people don’t spread your precious secret – Assume our 64 people only tell 48 new people about your secret

The spread factor will be and we will assume this remains the same

Calculate how many people know your secret every hour

75.064

48

6411275.06464

14875.075.06475.06464 17575.06475.06475.06464 32

Look at how many MORE people get to know your secret each hour…

64 112 148 175

48 36 27

Each hour there are fewer new people learning your secret and we can therefore predict that eventually nobody new will learn your secret

Draw a graph of ‘Hours’ (x-axis) versus ‘Number of People who Know your Secret’ (y-axis)

Take care when scaling your axes – make sure you know what numbers you need to plot

Fully label your graph (axes, titles, etc)

What does the graph look like – what shape is it?

Assuming the spread factor is less than 1, in other words the spread of your secret is reducing, there is a formula that we can use to calculate how many people in total will learn your secret…

If we call the number of people who already know your secret A and the spread factor is S, then

S

A

1secretyour hear whoTotal

How many people in total would learn your secret if 64 people already know your secret when the spread factor becomes steady at 0.75?

75.01

64secretyour hear whoTotal

256

Use the formula…

S

A

1secretyour hear whoTotal

to calculate the number of people who will get to know your secret if, out of the 200 people, one person in every ten tells another person

1.01

200secretyour hear whoTotal

people222

10

1Factor Spread

200A

Clearly, when the spread factor is small, fewer new people get to know your secret

Diseases Spread Like SecretsDiseases Spread Like SecretsIf the spread factor is less than 1, the disease will die outIf the spread factor is greater than 1, the disease will grow and spread

Typical Infectious Period

Spread Factor

HIV 4 years 3

Smallpox 25 days 4

Flu 5 days 4

Measles 14 days 17

In other words, at the start of an outbreak, a person with flu may be infectious for about 5 days during which time they will infect about 4 people

Investigation:

Imagine you have £1 and you put it in a bank account that pays 100% interest every year (wow!)

How much do you have at the end of one year?

Now imagine a different bank that pays 50% interest every 6 months - how much would your £1 become at the end of one year?

If a bank pays 25% interest every 3 months how much would your £1 become after one year?

If a bank pays 12.5% interest every 1½ months how much would your £1 become after one year?

Continue with this process… what do you notice is happening to your end of year total?

Initial Amount (£)

%age Interest

Multiplier

Number of

Payments per Year

EquationEnd of Year

Value (£)

£1 100% 1

£1 50% 1.5 2

£1 25% 4

£1 12.5% 8

£1 6.25% 16

£1 3.125% 32

£1 1.5625% 64

£1 0.78125% 128

£1 0.390625% 256

£10.1953125

%512

Initial Amount (£)

%age Interest

Multiplier

Number of

Payments per Year

EquationEnd of Year

Value (£)

£1 100% 2 1 1x21 £2

£1 50% 1.5 2 1x1.52 £2.25

£1 25% 1.25 4 1x1.254 £2.44

£1 12.5% 1.125 8 1x1.1258 £2.57

£1 6.25% 1.0625 16 1x1.062516 £2.64

£1 3.125% 1.03125 32 1x1.0312532 £2.68

£1 1.5625% 1.015625 64 1x1.01562564 £2.70

£1 0.78125% 1.0078125 128 1x1.0078125128 £2.71

£1 0.390625% 1.00390625 2561x1.0039062525

6 £2.71

£10.1953125

%1.00195312

5512

1x1.0019531255

12 £2.72

Investigation - Summary:

Although it initially appears that your £1 can be turned into an ever increasing amount by reducing the interest but paying more often it becomes clear that there is a limit to how much your £1 can be worth after one year

This value is £2.72 (rounded to the nearest penny)

The actual value is 2.71828  (5 decimal places)

This number (Euler’s number) occurs whenever you have natural growths in populations and can be calculated using the formula… n

ne

11

Euler’s Number (2.718):

If the number of people infected at the start of an outbreak of a disease such as Swine Flu is A and the spread factor is S we could create a very simple formula…

SAInfections NewBut newly infected people start infecting other people thus creating their own set of ‘New Infections’

Euler’s number is instrumental in predicting the total number of disease carriers during an outbreak of disease such as Swine Flu

The following formula has been found to work as a good model for predicting early stage infections…

1Period Infectious 1after Carriers ofNumber SAe

If 10 people have a variety of flu which is infectious for 5 days with a spread factor of 4, how many carriers of the disease will there be after the first 5 days?

1Period Infectious 1after Carriers ofNumber SAe14718.210

3718.210person) olenearest wh to(rounded 201

How many carriers of the disease will there be after the first 10 days (i.e.) after 2 infectious periods?

14718.2201 3718.2201

person) olenearest wh to(rounded 4036

How many carriers of the disease will there be after the first 15 days (i.e.) after 3 infectious periods?

1Period Infectious 1after Carriers ofNumber SAe14718.24036

3718.24036person) olenearest wh to(rounded 81040

How many carriers of the disease will there be after the first 20 days (i.e.) after 4 infectious periods?

14718.281040 3718.281040

person) olenearest wh to(rounded 226,627,1

If the formula for calculating the number of carriers after 1 period is…

1Period Infectious 1after Carriers ofNumber SAe

What would be the formula for calculating the number of carriers after T infection periods?

Period Infectious Tafter Carriers ofNumber TSeA 1

Use this formula to calculate the number of carriers after the first 6 infectious periods (i.e. about 1 month)

TSeA 1Period Infectious Tafter Carriers ofNumber

614718.210 SF) (3 000,000,655

Clearly this last figure is nonsense, this is because the formula works as a good model during the early stages of the spread of the disease but becomes inaccurate over longer periods

What factors prevent the numbers getting as high as 655 million?

…Many people infected so fewer people left to become infected

…Those remaining uninfected may already be immune to the disease

…Medication and drugs may have had enough time to start being effective

…People may be quarantining themselves as a form of protection

Typical Infectious Period

Spread Factor

HIV 4 years 3

Smallpox 25 days 4

Flu 5 days 4

Measles 14 days 17

TSeA 1Period Infectious Tafter Carriers ofNumber

Teacher’s Notes:

)1(

)1( n terms with Series Geometric Sum

r

raS

n

n

)1(

)1( n terms with Series Geometric Sum

r

raS

n

n

)1(

)1( n terms with Series Geometric Sum

S

SAP

n

n