Batch 3 - Class 226 - Viruses and Math

• Imagine an infinite chessboard that contains a positive integer in each square. If the value in each square is equal to the average of its for neighbors to the north, south, west and east, what should all the values look like?
• Answer: They are all equal. Proof by contradiction: assume that for any square, its value is different from one of its neighbors. Then there must exist a neighbor which has lower value. Now think about that neighbor - since the original square was larger, this neighbor must not have a neighbor which is even smaller. Thus, this chain must lead to a scenario where the numbers start to turn negative. Since this is not possible, all numbers must be equal.

• What are some of the key variables in spread of viruses
• How long does the person who gets infected remain contagious
• How long does she not display the symptoms (means that we can't isolate her)
• How many people can they infect on an average
• How many people do they come in touch with
• What are the chances of a person they come in touch with to get infected
• Ask students to think about the impact of each of these. How does the spread behavior change with change in each variable?
• Lets test these on a simulation
• https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Pandemics-How-Are-Viruses-Spread/
• Change size of population to large
• Now alter the different variables and see how the curve grows. Compare cases on two different windows to see how different parameters affect it - things to look for are ultimately what percentage of population gets affected, how quickly does that happen.
• Why is social distancing important: Reduces the number of people each person is in touch with. 
• Why wash hands more: Reduces the probability of transmission.
• What other factors could influence the chances of reducing the spread?
• What is the nature of growth - exponential, but we will come back to it
• Why do we care about slowing down in that case?
• Healthcare facilities - suppose 20% of people infected on any given day need hospitalization. See the length of the "infected" bar" to estimate beds required.
• Can you change the parameters so that at any point of time, not more than 10% people are infected?
• Understanding exponential growth
• What is linear growth (illustrate with a graph and growth per day)
• What is exponential growth - ask kids for an example (illustrate with graph and growth per day)
• What can lead to exponential growth? Why do viruses spread exponentially?
• What causes new cases are the old cases, so if the number of cases grow the number of additions next day will grow, and that forms an even larger base for the cases the next day and so on
• Can you think about other cases of potential exponential growth
• For example, population growth. https://qphs.fs.quoracdn.net/main-qimg-73e91afdafe2e2e46bf61aa62f32c110-c

• One can see this more clearly if one puts the y-axis on log scale (illustrate for 1,2,4,8,16 sequence) - what it means is that it may take 1 unit of time to go from 1 to 1024, and then in another unit of time, we are suddenly at a million. And so on.
• What does this mean if you see one country with a 1000 cases (India, currently) and another country with 100,000 cases (US)? Are we doing a 100 times better? Or just 30 days behind?
• So what happens as exponential growth keeps happening?
• Ultimately, most people are infected and hence the growth slows down - refer back to our simulation. Better still, we assume that people who recover from being infected develop immunity
• What this means is number of people being contacted per day should just be downscaled by number of people already affected. This gives a "logistics curve". It is undistinguishable from exponential in the beginning but then turns back
• How do we know when we are turning around?
• Inflection point. The point where number of new cases each day stops increasing and starts to reduce
• Lets spot these inflection points on our simulation
• In reality, the whole population may never be infected, the curve will start to turn before that. How do you detect that, and at that point, how can you forecast the infection trend? About 2x of where you are now.
• People are not really shuffled around the society. They are in communities. Should the growth still be exponential?
• Think of each community as a population with exponential growth
• Travel between the communities keeps the infection spreading between communities, so in effect we get an exponential curve even with very little travel.
• Hence, the travel bans
• See  https://www.youtube.com/watch?reload=9&v=Kas0tIxDvrg

• (MC Chap 0, Prob 1) - There is a jar of yeast and the yeast doubles every day. It fills the jar on 30th day. When was the jar half full?
• Answer - Day 29