Funnel plots of multiplying rates of exponential progress of Covid-19

Introduction

Pandemics and epidemics grow rapidly and are characterized by cumulative incidences that grow in an exponential form. Doubling times, i.e. the time intervals in which these incidences double, are often used as a measure of the growth1. In this paper, we use instead a more general form of a multiplying rate over a fixed interval to characterise the growth. We see that metrics related to Covid-19 from a large cohort of countries, or for the cohort of states and union territories with India, with varying population sizes, population age structures, quality of health care facilities, per capita income, reproduction numbers, etc. are dispersed within upper and lower bounds.  Funnel plots have become a standard graphical methodology to identify outliers when plotting summary statistics along with upper and lower bounds2,3. It is seen that the scatter plot of multiplying rates with deaths per capita yields a funnel plot from which outliers can be easily identified.. 

Materials and Methods

We start with an exponential model relating the cumulative deaths per capita N and time t:

                                                             N  =  N(0) e^mt                                                                  (1)

where  N(0)  js the number of deaths per million (or per hundred thousand) at an instant  t(0)    and   m   is a multiplying rate.  Note that     m   has the inverse units of t; hence if t is measured in days, m has the units of days-1. The conventional approach in characterizing cumulative incidences that grow in an exponential form is to use what are called doubling times1. These estimate the time intervals in which these incidences double and are used as a measure of the growth. Thus, if from t(1)to t(2), incidences double from N(1) to N(2), we have from Eqn. 1 that

N(2)/N(1) = e^m(t(2)-t(1))                                                           (2)

Taking natural logarithms, we get

m = ln 2/T(d)                                                             (3)

where T(d)  =  t(2) – t(1) is computed as the doubling time. That is,  T(d)   is a proxy for m.   

However, in this paper, we use instead a more general form of the multiplying rate over a pre-fixed interval T =  t(2) – t(1)   to characterise the growth: 

mT   =   ln(N(2)/N(1))                                                         (4)

We shall see that the scatter plot of multiplying rates m  with deaths per capita N(1) at  t(1)  yield a funnel plot. 

In our model, it is not important to us what  t(0)   or  N(0) are.  We are not trying to fit the entire curve, which needs the logistical equation to cover the peaking and declining phases. Instead at each instant, from t(1) to t(2), we assume that we are in an exponential growth phase and so we try to fit an exponentially based finite difference stencil. In this sense, it follows exactly the same logic as the doubling rate approach. In the latter, N(2)/N(1) is fixed at a value of 2 and t(2)-t(1) is calculated. The novelty here is that we keep t(2)-t(1) fixed and try to find N(2)/N(1), or rather, ln(N(2)/N(1)).

Our World in Data maintains a collection of the COVID-19 data which is updated daily and includes data on confirmed cases, deaths, and testing, as well as other variables of potential interest:

 HYPERLINK "https://github.com/owid/covid-19-data/tree/master/public/data"

We downloaded the death rates per million of population on 01/07/2020 (t1())  and 31/07/2020 (t(2)), for all countries in the data set (accessed on 01 August 2020).   Thus, we are using the most recent 30-day interval to compute a multiplying rate for each country. Shorter or long periods can also be used. For each country, we compute  N(1), N(2), N(2)/N(1), and mT, using the expressions in Eqns 1 to 4. 

MyGov gives the Covid-19 state-wise status in India:

 HYPERLINK "https://www.mygov.in/corona-data/covid19-statewise-status"

We downloaded the deaths and computed the death rates per 100000 of population on 15/07/2020 (t1)  and 06/08/2020 (t2), for all states and union territories in the data set. This corresponds to the most recent 23-day interval. The multiplying rate for each state or union is computed in the same manner as before using the expressions in Eqns 1 to 4. 

Results and discussion

Table 1 shows the multiplying rates over a 30-day period for countries whose population exceed 50 million.

Figure 1 shows the dispersion of the 30-day multiplying rate vs the death rates per million of population on 01/07/2020. What is interesting is that the numbers then fall neatly into the inverted funnel plot. It gives us an idea of where a country is within bounds. Ethiopia and Kenya appear as outliers. South Africa and Colombia are found right at the top of the inverted funnel. At the very base are countries like the United Kingdom, Italy and France which have high death rates per million, and countries like Myanmar and Tanzania which have very low death rates per million. These form respectively, upper and lower bounds of the funnel plot. China, South Korea and Japan have death rates which are a fraction of that of countries in Western Europe but have very low multiplying rates. India, Indonesia and Bangladesh are large countries which have threateningly large multiplying rates and are clearly in a danger zone. India is right at the middle now and like Belgium, etc. should move to that corner. It looks like it will take another 100 days. Krygyzstan's data seems to be very erratic!

In the early days of the pandemic, most of the reported deaths came from the rich countries and the poorer countries seemed to have been spared5. However, we now see that with the passage of time many of these countries are appearing as outliers. Figure 2 shows the funnel plot depicting the dispersion of the 30-day multiplying factor vs the death rates per million of population on 01/07/2020 for the outlier countries (solid circles) against the backdrop of the rest. The multiplying rates over a 30-day period for outlier countries are given in Table 2.

Table 3 shows the multiplying rates over a 23-day period for states and union territories in India.

Figure 3 displays the dispersion of the 23-day multiplying factor vs the death rates per 100000 of population on 15/07/2020 for these states and union territories. Again, the numbers fall neatly into an inverted funnel plot.

Delhi, which seems to have passed the peak, has reached the regime of low multiplying rates and high death rates and occupies the bottom-right hand corner. The countries not shown, and which should occupy the vacated space at the bottom-left hand corner are the states which have had zero deaths at the beginning of the window. These are: Andaman and Nicobar Islands, Dadra and Nagar Haveli and Daman and Diu, Manipur, Mizoram and Nagaland. This is because zero values cannot be plotted on a logarithmic scale. Arunachal Pradesh is conspicuous in that it has not registered a change during these 22 days. If we consider Tripura to be an egregious outlier, then Ladakh shows the highest multiplication rate during this period.

Concluding remarks

Using data available in the public domain we showed that over the most recent 30-day period, COVID-19 death rates are related to the death rates in a manner that can be captured by what are called funnel plots. Instead of doubling times to capture the nature of the exponential growth, we have used what are called multiplying rates. Funnel plots allow us to identify the outliers very easily. It is seen that many regions which seemed to have been spared in the early days of the epidemic are now seeing multiplying rates which are higher than expected of those lying within the bounds.

Acknowledgements

The author benefitted from discussions with Dr Ajit Haridas and Dr P Jafarali.