Math of covid-19

The rate at which microbes multiply is proportional to the microbes present at a given time, provided other factors like nutrients and temperature remain favorable.

\frac{dN}{dt} = k N

This equation has the exponential solution N(t) = N_0 e^{kt}. Spread of infections can also be modeled along similar lines, since the rate at which people become infected is expected to be proportional to people already infected.

Interestingly Nicolas Vandevalle has made a model to fit the covid-19 data for Belgium to an exponential curve.

As can be seen, the initial cases of infection follow an exponential curve to a very good approximation, however as the effect of social distancing comes into play, it is possible that the curve may become linear. The data was analyzed for Belgium. Right now, the data is insufficient to say if the curve is turning linear, but the implications are far-reaching.

A Mathematical Scam

Monetary scams by their nature are mathematical. Here is a story of an interesting one. Suppose you receive a mail from a company saying that share prices of X is going to go up the next day and offers services to help you invest, you being a wise person ignore the tip and find to your mild amazement that the prices actually did go up the next day. A week later you the whole process repeats and now you are a little more than mildly amazed that the prediction came true. Suppose this process goes on for five weeks, if you are not well trained in thinking rationally, you now trust the predictions to a large degree and contact the company for a decent sized investment.

Here is what is quite likely to be going on. The scammers send similar mails to say 100,000 people with randomized predictions of say 10 different types of shares going up. Out of which let us say 50,000 predictions came out to be true. The next mail goes only to those 50,000 and so on. At the end of the fifth round about 6250 of those brave-hearts who have not given-in to the desire of making big money are quite likely to do so now.

So while managing your money, if something seems too good to be true, your hunch may be correct.

Reading bicycle tracks

You might have come across real life accounts of how a seasoned hunter can read the pug-marks of animals and figure out surprising amount of information from them, quite Sherlock Holmes style. Now move this scenario to the concrete jungle and consider the case where you are given track marks of a bicycle wheels, can you figure out which way was the bicycle moving?

In the book “Genius at Play”, a biography of John Conway, Siobhan Roberts discusses an interesting problem. For one of the classes, Conway and his co-teachers rode bicycles on large rolls of paper and handed out the resulting tracks to the students to analyze and figure out which way the bicycle moved.

There is an easy algorithm to answer the question, which has to do with curvature of the tracks and drawing tangents and measuring distances along them.

Notice that the track made by the front wheel always has a larger curvature because when a turn is being made by the front wheel, the back-wheel is free to move only along the direction of the bicycle.

If you draw a tangent to a point P on the track made by the back-wheel of the bicycle, then the direction of the tangent shows the direction of the velocity of the back-wheel. Since this velocity is along the direction of the bicycle frame, if one moves along the tangent, there has to be a point on the outer (front-wheel) track which is exactly at the distance between the two contact points of the wheel. Thus the direction along the tangent in which the distance between the contact point P (on the inner curve) and the point on the outer curve Q stay the same, no matter which point P you choose, is the direction of motion of the bicycle.

Agony and Ecstasy of Mathematics

There are two kinds of people in the world, those who enjoy mathematics and those who don’t. What is interesting is that even those who enjoy mathematics don’t quite agree on how should mathematics be taught. On one side there are people like the famous mathematician John Conway who would carry fruits and vegetables to the classroom to teach students about geometry and curvature while on the other side there are people like the famous computer scientist Dijkstra who insist that right from the start, students should learn rigorous abstract concepts. My own belief is that majority of people develop appreciation for mathematics only through specific examples and that it takes a level of expertise before which one may start appreciating rigorous mathematics. I belong to the first category of people who think that playing games, looking at bicycle tracks and thinking of real life situations is a much better way of getting introduction to mathematics.