So you're stuck at home with the kids. Why don't you and/or your kids learn some math with this *long* thread on exponential growth? I'll try to make it accessible given twitter limitations, but feel free to ask questions! Exponential growth today and epidemic models tomorrow.
Let’s start by learning about exponential growth. It's very important to understanding our predicament.
Both the cumulative (total) number of COVID-19 cases and the number of new cases are growing exponentially. What does that mean?
Both the cumulative (total) number of COVID-19 cases and the number of new cases are growing exponentially. What does that mean?
Adding fixed amount per unit time is linear growth. Multiplying by a fixed amount per unit time is exponential growth. If one case multiples into 2, and 2 into 4, over some time t, then exponential growth. In discrete time units, like days, we often call this “geometric growth”
Suppose you have some number of cases at the current time, t=0. Let’s call that number of cases N0. We can write the number of cases in the next time step, t=1, as N1, which is proportional to N0 by a constant that we often call “lambda” in population models, or symbolically λ. 
If λ=1, then the number of cases stays the same. If λ>1, then the number of cases grows. λ<1 is not possible when modeling cumulative cases. One feature of exponential growth is a constant doubling time. If cases are currently doubling every 3 or 4 days, how can we determine λ?
As of March 16 we have 4645 cases in USA, but the true number is much higher. Let’s assume that we’ve only missed half the cases so that we currently have about 10000 cases. Let’s see how cases will grow over the next 30 days given a 3 vs 4 day doubling time.
OK. Right about now it should click that *this* is why scientists and public health officials are freaking out and shutting everything down. If these doubling times continue and we did nothing, we could easily have millions of cases in a month! But this can't go on forever, why?
Eventually the number of susceptible hosts (jargon for humans without infection) becomes limiting and Infected people increasingly interact with recovered people who have immunity and cases start to decline. This will slow exponential growth but ONLY once large fraction infected
We very often see the COVID-19 cases plotted through time on a logarithmic axis. Why? Let’s find out by starting from our growth equation.
Say we plot log(cases) through time and find out from the slope that λ=1.33. This is a 33% growth rate, which we calculate as (λ−1)∗100. How do we determine the doubling time? Well, we need to go from N0 cases to Nt=2N0 cases. So...
OK. Now you see that we can directly relate the slope of this plot to the doubling time. So what do the data actually tell us and how does it vary by country? Perhaps you've seen plots like this one?
This is a holy shit moment. With a 2-3 day doubling time, we could have MANY MORE THAN 10 MILLION CASES IN A MONTH IF WE DON'T DO SOMETHING! 10 million is not enough to appreciably reduce the pool of susceptibles given 327 million people, so the exponential approximation is ok.
10 million cases in a month is not a future we want. How do we avoid this? Well, we have to get new cases to stop growing. We have to bring lambda to 1 through social distancing, contact tracing, and testing. Otherwise we lost. Tomorrow we'll project with a real epidemic model.
This model has been in discrete time. In continuous time we start with a rate of change of cases per time that is proportional to the current number of cases. dN/dt=rN. The solution to this "differential equation" is Nt=N0*e^(r*t). "e" is the exponential in "exponential growth"
You can easily go between exponential and geometric growth if you notice that e^r is the same as λ, so the equations are really equivalent. OK, that's about it for today.
