### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

# Epidemic Modelling

##### Stage: 4 and 5 Challenge Level:

The following solution was sent in by Thomas from Dalton Primary School, New York. If you repeated Thomas's experiment with the same simulation parameters you would get different results. Can you think why? It is because the results depend on probabilities. To get reliable results that we can base decisions on we need to find the average (or mean) results from many repetitions of the same experiment with exactly the same parameters.

Thomas's results are interesting because they show very different outcomes according to whether the sick people circulate in the village, or stay at home or are put in total isolation. Lewis from Highcliff Primary School also says that isolation is a good policy but when do you think it is advisable and why?

#### Thomas's results

I modelled a large village being affected by a very lethal and infectious disease and looked at the impact of mobility and isolation on the length of the epidemic, the number of deaths, the number of infections, and the number of recoveries.

 Mobility Duration Deaths Not infected Recovered Normal 169 342 33 125 Static 35 15 481 4 Isolated 9 1 498 1

#### Conclusions:

1. Reduced mobility and isolation had an enormous impact on the duration of the epidemic, the number of deaths, and the infection rate.
2. Isolation was more effective than people remaining static when infected.
3. This suggests that when there is a dangerous epidemic (High infection rate, high death rate), effective public health policies would be to tell people to stay at home and to isolate those who are sick.

Ruth from Manchester High School for Girls sent us this careful investigation of a different aspect of this problem. She repeated each experiment several times and drew conclusions from the mean of several runs.

#### Ruth's Results:

I am investigating whether the incubation period of an illness affects how useful it is to isolate infected individuals.

 The simulation parameters were: Grid Dimension 25 Initial Population 150 Initially infected 25 Initially immune 0 Days ill 8 Probability of death 0.9 Probability of infection 0.9 Probability of static 0.1 Gain immunity true

Independent Variables:
Days before infectious
Behaviour if ill

#### RESULTS

Normal when ill \begin{array}{lllll} & \text{Duration} & \text{Deaths} & \text{Never Ill} & \text{Recovered} \\ \text{Mean} &19.8& 135 &0.2& 14.8\\ \text{St. dev.} &2.7& 3.6& 0.4& 3.5 \end{array}

If the behaviour when ill is normal, the number of days before infectiousness makes no difference.

#### Isolated when ill

0 days before infectious \begin{array}{lllll} & \text{Duration} & \text{Deaths} & \text{Never Ill} & \text{Recovered} \\ \text{Mean} &9 &21.6& 125& 3.4\\ \text{St. dev.} &0&1.5 &0 &1.5 \end{array}

1 days before infectious \begin{array}{lllll} & \text{Duration} & \text{Deaths} & \text{Never Ill} & \text{Recovered} \\ \text{Mean} & 15.2 & 81.8 & 56.8 & 11.4 \\ \text{St. dev.} & 1.6 & 5.5 & 8.4 & 3.4 \end{array}

2 days before infectious \begin{array}{lllll} & \text{Duration} & \text{Deaths} & \text{Never Ill} & \text{Recovered} \\ \text{Mean} & 18.6 & 113.4 & 26.8 & 9.8 \\ \text{St. dev.} & 4.4 & 7.2 & 6.4 & 4.6 \end{array}

3 days before infectious \begin{array}{lllll} & \text{Duration} & \text{Deaths}& \text{Never Ill} &\text{Recovered}\\ \text{Mean}& 19 &126.2& 10.8 &13 \\ \text{St. dev.} & 1.5& 6.3 &7.5 &1.7 \end{array}

4 days before infectious \begin{array}{lllll} &\text{Duration}& \text{Deaths}& \text{Never Ill} &\text{Recovered} \\ \text{Mean} & 19.2 &129.2 &4.4 &16.4 \\ \text{St. dev.} &1.9 &4.1& 2.2& 4.2 \end{array}

5 days before infectious \begin{array}{lllll} &\text{Duration} &\text{Deaths} &\text{Never Ill}& \text{Recovered} \\ \text{Mean} &18.8 &133 &0.6 &16.4\\ \text{St. dev.} &1.9 &2.8 &0.8 &2.9 \end{array}

6 days before infectious \begin{array}{lllll} &\text{Duration}& \text{Deaths}& \text{Never Ill} &\text{Recovered} \\ \text{Mean} &21.8& 133.2 &0.6& 16.2 \\ \text{St. dev.}& 4.7 &4.2 &0.8 &4.5 \end{array}

7 days before infectious \begin{array}{lllll} &\text{Duration} &\text{Deaths}& \text{Never Ill}& \text{Recovered} \\ \text{Mean}& 20& 133& 0.4& 15.8\\ \text{St. dev.} &1.7& 4.1 &0.8 &3.3 \end{array}

8 days before infectious \begin{array}{llll} &\text{Duration} &\text{Deaths} &\text{Never Ill} &\text{Recovered} \\ \text{Mean}& 19.6 &136.8& 0.2 &13 \\ \text{St. dev.} &0.8 &1.7 &0.4 &1.7 \end{array}

This disease is very lethal and infectious. If nothing is done, it will kill most of the population of the town. Isolation is an effective way of reducing the death toll and duration of the epidemic.

If the time before infectiousness is a large proportion of the duration of the illness, it makes very little difference to the outcome whether or not infected people are isolated. The percentage difference between the number of deaths when isolated and not isolated is under 1.5% when the period before infectiousness is over half the length of the illness (5 days or more) but is over 80% when the period before infectiousness is 0 days and is almost 40% when it is 1 day. The variation in the length of the epidemic follows a similar pattern with less than 1% variation between isolation and non-isolation for 7 or 8 days before infectiousness but over 50% difference for 0 days and almost 20% for 1 day.

These results show that, while isolating infected individuals will almost always reduce the death toll and end the epidemic sooner, it is most effective when the incubation period of the illness is relatively short. As the incubation period increases, the amount of time that infected individuals are isolated for, and therefore the amount of time they are not infecting others for, decreases, so it is not unexpected that this is the case. The results suggest that, as isolating infected people would be quite difficult and expensive, it is only worth doing so if the incubation period of the infection, when they are infectious but show no symptoms, is quite short compared to the period when they do show symptoms so would be isolated.