Working on herd immunity

Having been working out R numbers from the government data, I was wondering about how they came up with figures for herd immunity. This is when so many people have had a virus and have antibodies that stop them getting infected again that anyone with the virus will struggle to find anyone to infect. I'd heard originally that the models for Covid 19 suggested about 60% but saw an article in The Times newspaper at the end of last month that suggested that under current conditions that figure could be as low as 15%. I was struggling to work out my own theory but I found this wonderful explanation: https://www.quantamagazine.org/flu-vaccines-and-the-math-of-herd-immunity-20180205/ The article gives an excellent account for how to calculate the percentage for herd immunity. The formula = 1 - 1/Ro. I plotted the graph

I was confused because the R value is below 1 in the UK at the moment but the graph won't go below 1. Then it became clear that there is more than one R number. Ro is the number for an unhindered virus with no social distancing or other controls. I believe that the modellers in March were using a value of Ro for Covid 19 of 2.4, hence the 60% figure. That outs it much less contaguious than measles, which is quoted as Ro = 12.

Year 10 Air resistance experiment 2: reproducible

This experiment investigates the same theory as the parachutes experiment. It will have the same independent, dependent and control variables.
Get a sheet of A4 paper - use a page from your book if necessary. Fold it into quarters

 Carefully tear or cut the four quarters.

 Take the first piece, fold it in half longways and draw a line 1 cm up from the open edge.

 Fold the corners down to the line.

 Fold the bottom edge up along the line.

 Repeat on the other side.

 Open it out to make a tiny paper hat.

Here's why I am referring to them as "hats".

 Hold it up as high as you can with the open side downwards to catch the air. Let it fall and time how long it takes to hit the ground. Do this 3 times to get repeat readings for a mean.

 Make another one and slide it into the first one.

 You now have a double "hat". Open it out. It now has twice the weight but the same area for catching the air because one is sat inside the other.

Hold it up as high as you can and do 3 repeat readings. Do the whole experiment with 3 stacked inside each other and then 4. The table is below.

Here is the graph of my results

We had the same independent variable as the parachute experiment, the same dependent variable and the same control variable, even though it was a different experiment. The graph is the same shape showing the same pattern in the results. When a different experiment investigates the same variables but gets the same pattern in the results, we say that the pattern is REPRODUCIBLE. It is extra evidence that the theory about air resistance and terminal velocity is correct.

Year 10 Air resistance experiment 1

I got a plastic sandwich bag from the kitchen and taped a piece of string to it. I made a loop in the string so I could hang paperclips from it.

Then I held it up as high as I could, making sure the bag was open as much as possible to catch the air. I timed how long it took to fall to the floor.

I started with no paper clips on the string at all. When an object is falling in the air, the weight (gravitational attraction) is pulling it downwards as its driving force. The air resistance acts as a counter force to try to stop this happening. Air resistance depends on both area and speed. As the bag falls, the weight makes it go faster. This increases the air resistance. When the air resistance becomes as big as the weight, the speed stops increasing. It carries on down at steady speed which is called the TERMINAL VELOCITY. When I started adding paperclips the weight got bigger so it had to fall further and was able to go faster before it reached terminal velocity. So adding more paperclips makes the time for the fall decrease as shown on the graph of my results below. Notice that it is levelling off. If there was no air in the room, it would have taken 0.66 seconds to hit the ground.

 The results table for the experiment is below.

The number of paperclips was the independent variable because it was the thing I deliberately chose to change. It is the independent variable because I can write in the numbers on the table before I even start.
The time is the dependent variable because these are numbers that I can only find out by doing the experiment.
I kept the height of fall the same throughout the experiment so that was a CONTROL VARIABLE.

Covid data update - a new gradient

I added a weekly point to my log graph of the Covid data and the last 3 points form a new stright line of higher gradient. Putting this into my model where I had the gradient being (R-1)/T, with T being the time for the next set of people to become infected, I calculate that if R was in the range 0.7 to 0.9 before it would be in the range 0.52 - 0.84 now.

Year 10 rolling tin distance-time graph experiment

For this experiment you need a board or a hard-backed book. You need to prop up your book. I propped it up on an exercise step. Then I measured 50 cm along the floor from the bottom of the book and put string on the floor to mark the finishing point. You don't have to use string - you could mark it with something else.

 I got a full food tin and released it from the top of the ramp so that it could roll down.

I started my stopwatch when the tin reached the floor at the bottom of the ramp. I was timing the tin as it rolled along the floor towards the string.

I got into position so that I could see exactly when the middle of the tin crossed the string. I stopped my stopwatch when this happened. If I hadn't got in position to look down on the tin crossing the string there would have been a parallax error.

I did two more repeat readings for 50 cm and calculated the mean. Then I moved the string to 75 cm from the bottom of the book and did the experiment. Then the string moved to 100 cm and so on. Each time I was deliberately changing the distance that the tin was rolling. Here is what the results table looks like:

This is what my distance-time graph looked like:

You can tell that the tin slowed down because the line startes steep and then is less steep at the end. The gradient of a distance-time graph tells you the speed.

Moments experiment

For this experiment you need a pen or pencil, a ruler and some sweets. I used jelly babies. Starbursts would be good but other brands are available... Balance the ruler on the pencil.

Then put a sweet 10 cm to the left of the pencil.

 This has made the ruler rotate anti-clockwise. The pencil is acting as a pivot. The weight of the sweet acting at a distance away from the pivot creates this turning effect. This turning effect is called a MOMENT. Moment = force x distance from pivot. In this case the force is the weight of the sweet. We are going to say that the weight of the sweet = 1, because we have 1 sweet.

 Now add a sweet on the right hand side. This creates a clockwise moment. Find the distance from the pivot where the ruler balances as best you can. It is very hard to get perfect balance but you can get a feel for the right point.

Next leave the single sweet on the left but balance it with two sweets on the right.

Put your results in a table like this:

In theory, for balance, weight x distance on left = weight x distance on right.
Your experiment will not have been perfect but should be close. I'll be asking you how close.
Finally, what we have just found is called The Principle of Moments
It says "For equilibrium, total anti-clockwise moments = total clockwise moments".

More calculations with my Covid data model

I found more tests that help me to work on my value for T in the equation I came up with for the Covid data. T is the time it takes for one person to pass on the virus. Professor Neil Ferguson said the cases were doubling every 3 - 4 days before lockdown. We know that the R rate was roughly 3 at that point. Here's how you do the doubling analysis.

 Putting in t=3 days fpr doubling gives a value for T of 8.6 days. When t=4 days, T = 11.5 days. Both values would sit inside the 14 day quarantine. There was more data about regional R values in the Daily Mirror this week. https://www.mirror.co.uk/news/uk-news/coronavirus-r-close-tipping-point-22186919 It gave ranges of R values with a mean along with the time for cases to halve. Here's the maths for halving.

I put in the mean values with the halving times and got values for T of between 3.5 days and 5.2 days - shorter times than the doubling analysis. I was also interested to note that when R=1 there is a halving time. This must be rounded because when R=1 the numbers remain the same. I estimate they rounded from R=0.97

Update on the Covid data graph

My graph of the Covid data for 7 day rolling average of UK deaths has data points every Wednesday so I have been able to update it today. After 2 months this is the first sign of a significant downward deviation from the straight line. If the trend based on only two data points continues then by next week the 7 day rolling average should be down to 106. I am coming to the conclusion that it is the job of science to attempt to predict the future.

Developing a simple mathematical model for the Covid data

Having seen my straight line log graph for the Covid data, I set out to develop a mathematical model. The famous R number is the number of people that one person infects. So if the R number is 3 and there are 5 people with Covid, then each will infect 3 and the number infected by them all is 3 x 5 =15. This means that at each stage the number infected = R x N, the number at the previous stage. To find the increase in number you do RN - N, the number at the end - number at the beginning of each stage. In the picture below I developed that into dN/dt, the rate of change of the number. I then integrated to write N in terms of t. There's a mistake in the last equation that I've just noticed. I've changed the N to a t in red.

Now the problem then becomes "what is t?" In the model above it is the stage in virus transmision from one person to the next. It isn't time in days. It's not a conventional concept of time. But all the data I have is measured in days - numbers of people on particular days. So I adapted the equation by inroducing a term T which is the number of days that it takes 1 person to infect R other people. Then t in my equation below is in days. I took natural logarithms and rearranged to get the y=mx+c form for a straight line graph.  

The gradient is (R-1)/T. Problem is that we don't know T. We do know that the gradient is -0.0282. We also know that nationally the R number is between 0.7 and 0.9. Putting R=0.7 into the equation gives T as 10.6 days. Putting R=0.8 gives T as 7.1 days. Putting R=0.9 gives T= 3.5 days. These values are consistent with given incubation times. I believe that it is methods like this that are used to calculate the R value and the different models used must have different values for T. I have found more data to use to test my model.

Looking at the Covid data

I have been listening to the data in the UK government daily Covid briefings and trying to get an understanding for the numbers that are given out. The number of positive cases is a difficult statistic to follow across the last 4 months because the amount of testing has increased so the number of cases found would have incraesed with it. Sadly the most reliable data has been the 7 day average of the daily deaths. I was wondering if the fall in numbers of deaths was expinential so I took the natural logarithm of that figure once a week. I got most of the numbers from a graph on this site https://coronavirus.data.gov.uk/ The data gives a straight line which means a clear exponential decay. I hope that the recent falls in hospital admissions will mean that there will be an abrupt downward change in my graph. Knowing that the relationship is exponential is helping me to develop a simple mathematical model.

Year 10 Inertia experiment

To do this experiment you need a plastic container, some tins of food, elastic bands and a ruler. Here are some possible containers. You'll see as we go on that I actually used an empty margarine tub.

You need to make a catapult around the legs of a chair using the elastic bands.

 One elastic band wasn't long enough so I had to loop three together. Poke one through another elastic band and pull it tight back through itself as shown below.

 Here's my band of three.

 I put my lightest tin of food into the tub and pulled it back as far as it would go. The red tin at the back was used to mark how far I pulled it back because to make it a fair test we need it to be the same each time. The elastic band provides a force and it is important for Newton's Laws that the force is a control variable (the same each time)

 I let go and the tub slid forward. Notice that I had to wedge the single tin with a glove to stop it moving around and dissipating energy when it was sliding. I measured how far it slid from my marker tin to the back of the margarine tub.

 I then doubled the mass by using two tins.

 I used different combinations of tins to get different masses. At each stage I did 3 repeat readings and got the same pattern of results for each repeat, even though the results were not always perfectly identical due to random error. By doing this I can say that my experiment is REPEATABLE.

 You can read the masses from the sides of the tins. My smaller sweetcorn tins were 150 grams and the heavier tins were 400 grams.

 I did the whole experiment again on a laminate floor and found that I got better results. The friction force is smaller. We say an experiment is REPRODUCIBLE if we get the same pattern of results by a different method or by a different person doing the original experiment. I'm going to argue that getting the same pattern on a different surface means the experiment is reproducible because even though the slide distances were different to the carpet, the graph looked the same shape. If you do the experiment and get the same pattern as I got then it will definitely be reproducible.

Particle flow of scree in Gable Gill on Longside

We went up Longside direct up Gable Gill from Dodd Wood. The scree at the top seemed to present an easier option than knee-deep heather. I'd be instinctively wary of the bigger block scree but this turned out to be more stable than the smaller scree. Two possible ideas for why:
1. Smaller particles can pass each other more easily. Each one has a smaller surface area and thus will experience less friction so will flow more easily.
2. The larger blocks are heavier and have more inertia. Hence each individual block needs a bigger force to get it to move.
It's probably a combination of the two factors. We did make it to the top and here's the view...