# Doubling time

(A detailed explanation of the method and inferences with respect to the lockdown can be found here.)

Mathematical models used to characterize early epidemic growth feature an exponential curve. This phase of exponential growth can be characterised by the doubling time. Doubling time is the time it takes for the number of infections to double. Our study here has been inspired by the study of Doubing times of COVID-19 cases worldwide by Deepayan Sarkar.

For a given day, the doubling time tells you the number of days passed since the number of cases was half of the current count. A higher doubling time means it is taking longer for the cases to double, and indicates that the infection is spreading slower. Conversely, a lower doubling time suggests faster spread of infection. For an infection growing at a constant exponential rate, the doubling time is constant. However, as observed in the COVID-19 situation, due to interventions like social distancing, lockdown and containment of hotspots of infection, the doubling time fluctuates and is a function of time. It also varies between districts, states, and countries which maybe in different stages of infection.

One technical note before we begin: All the graphs start from the first date on which the count surpassed 50 and the number of infections was at least twice as that on 10th March. Hence, different graphs have different starting dates.

### All India Doubling time

Consider the graph below, where we have plotted the all-India infection time line on log scale. The different phases of the lockdown have been colored differently and a regression line has been fitted for each phase of the lockdown. The slope of the regression line is the effective rate of exponential growth and effective doubling time is $\log(2)$ divided by the rate of growth. The legend indicates the time taken for infections to double in each phase.

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

As indicated above, the dots represent the number of infections over time on the log scale. The y-axis has been plotted in the log scale, that is, $10, 10^2, 10^3, ...$ will be equidistant instead of $10, 20, 30, ...$ on the linear scale.

In the second approach, one may track the doubling time on a daily basis. One can keep track for any given day how many days ago was the number of infections half of the current count. More precisely, for a given day on the x-axis, on the y-axis we plot the number of days in the past when the infection count was half of the given day.

Hovering over the dots will give you the precise value of doubling time. For example, if we hover the mouse over March 30th then we find that the doubling time is 5.04. Indeed, on March 30th the total number of infections was 1200 odd and March 25th the total number of infections was 560 odd.

One can compare the doubling times of India with other countries at the Doubing times of COVID-19 cases worldwide website.

To proceed on any further inference we must take into consideration the different rates of testing and quarantine measures in different states/countries. The graphs simply represent the number of positive cases which directly depends on the amount of testing. Hence one can't immediately conclude that places with higher doubling times are doing better without accounting for other aspects of this epidemic.

Using doubling time we can understand whether or not the growth of the infection is in the exponential phase or not.

We shall now examine graphs of Maharashtra and Kerala. For Maharashtra, in the log-scale, we observe

that the slopes of the regression lines for each phase of the lockdown specify the effective doubling time. In the doubling time graph we observe

that doubling times don't seem to follow any trend till April 27th, 2020. Since then they have been roughly increasing till 17 June. Thus one may understand the growth of the infections in Maharashtra by taking the two graphs together.

Next we examine Kerala. In the graph below, the doubling time has been increasing consistently till May 15th, 2020. However, from 22th May onward, the doubling times have been decreasing, with a sharp dip seen on 24th May, 2020. This can be attributed to the relaxation in lockdown which allowed migration into the state.

Below, in the log-scale graphs it can be observed that the different phases of the lockdown have different effective doubling times.

In the States Time Line Pages we have plotted the doubling time graphs for states with more than 90000 infections, for states that have between 90000 to 9000 infections and for states with less than 9000 infections. As with Kerala, other states also may show a decrease in doubling time after 22nd May due to relaxation in the lockdown.