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