# How to tell when exponential growth is ending?

Several measures such as quarantine, lockdown, increase in testing, etc. are being taken to arrest the spread of COVID-19. Hence one would like to judge if indeed there is a slowdown in the rate of growth of infection. To see this one may study the "first derivative" of the infections. In the below graph we have plotted the number of new infections, recoveries and deceased on each day.

It is easy to observe that if the infection is in the exponential growth phase then first derivative at any given time is proportional to current number of infections. From the "first derivative" graph above, one may observe that the daily new cases fluctuate quite a bit. This may make it harder to conclude if the infection is in the exponential growth phase.

Aatish Bhatia, in collaboration with Minute Physics, tries to understand the trajectory of the COVID-19 confirmed cases and deaths. They focus on whether the curve is flattening or equivalently, if infection growth is slower than exponential growth. To achieve this, they plot the total number of new infections in the past week versus the total infections up to that day. If the infections are following an exponential growth then the resulting trajectory of a country will be on "the straight line". When the growth stops being exponential, the curve will dip and its trajectory will fall below "the straight line". On their website they illustrate this via a dynamic interactive graph for many countries. There, one can observe how the trajectories of various countries are faring with respect to "the straight line". A detailed explanation of the above method can be found here.

We use the notion of moving average over 6 days to reduce the fluctuation of the first derivative. That is, we plot on each day the net increase from three days before that day to three days after that day on the $y$-axis and the total number of infections up to that day on the $x$-axis. Both the axes are on log-scale.

For all the graphs here as you hover your mouse pointer over the graph annotations with details will be displayed.

In the above plot the trajectory closely follows "the straight line" indicating that the infection is in the exponential growth phase. Next we plot the same for those states of India who have at least 150 confirmed cases as of yesterday.

If you click on a state in the legend, it's curve will disappear. On clicking on it again, its curve will reappear. To see the plot of only certain states you may click on the names of other states in the legend. To isolate one State's curve, double click on it's name in the legend.

Using the tools on the top left of the graph, you can zoom in/out and select specific parts of the graph. To reset the scaling, you may either autoscale [if you have selected a subset of the states to be shown] or use the Reset axes option. One can use the spike lines option to see dotted lines from the axes appear. To see multiple states data at once you can select Compare data on hover.

The different curves in the above graph correspond to different states. Each state's graph begins on the day when its cumulative infection count exceeded 20 cases. We observe that many state's trajectories seem to be roughly on "the straight line" and consequently we can conclude that the COVID-19 epidemic in these states is in the exponential growth phase.

In the below plot we examine three states: Karnataka, Kerala, Maharashtra.

We observe that all three state's trajectories are roughly on "the straight line" till March 26th, 2020. The graph of Kerala begins to deviate from this line from this date and fall of exponential growth. This trend continues until May 6th, 2020 with some fluctuations around April 15th, 2020. On May 6th, The net increase from three days before that day to three days after that day reaches its running minimum. After this date, it has started to increase again. The graph of Karnataka also seems to deviate from "the straight line" on 18th April, 2020 before starting to increase again on May 5th, 2020. Further, the graph of Maharashtra seems to be continuing on "the straight line". To understand the precise rate of growth of infections one would have to examine the structure of the graph in more detail.

For any further inference we must take into consideration the different rates of testing and quarantine measures in different states. The graphs are derived from information provided by the number of positive cases which directly depends on the amount of testing. Hence one can't immediately comment on the relative performance of states without accounting for other aspects of this epidemic.