When will the Covid-19 Pandemic End?Levitt’s Metric on Indian Data

When will the Covid-19 pandemic end? This paper seeks to answer this important question using the metric proposed by Michael Levitt in Mar 2020. We plot the metric for data corresponding to various Indian cities and states, and compare it with the plots for places like Italy and Spain, where the pandemic has “ended”. The projection is that Covid-19 deaths will become “insignicant”, by early Aug in Mumbai and Delhi. Similar projections can be made and are likely to be useful for other regions.


Introduction
The Covid-19 pandemic has brought much of the world to a stand-still. The question foremost on everyone's minds is "when will the pandemic end?". Michael Levitt, in his data analysis in Mar 2020 [1] of data from Hubei (China), proposed a simple metric, which shows a linear decreasing trend with time. In this paper, we apply this metric to data from various regions of India, to project as to when the pandemic is likely to "end".
When can we say that a pandemic has ended? In this paper, we say the disease has "ended" as a pandemic when the number of daily deaths falls to "insigni cant" levels (not necessarily zero), such as around 1/20th-1/30th of the daily non-Covid death rate.
Advantages: Aside from the simplicity of the metric, it has various other advantages. Since it is a ratio, it is independent of population size, and hence it can be used to compare across regions of any size: city, state, district, or country. It is robust to different countries/regions having different testing ca-pacities, or varying de nitions of what is a Covid death. It is also robust to systematic under-counting, so long as such under-counting is time invariant.
Deaths vs cases: In this paper, we compute the H(t) metric on Covid-19 deaths rather than cases. Number of cases as a metric is not good for a number of reasons. First, it depends on testing capacity which varies temporally. It also depends on people's access to tests, which in turns depends on many policies: are private tests allowed, are tests without prescription allowed, do people have transportation to go to test centre, etc. which are all time variant. To add to this, number of tests also depends on social acceptance of tests, which is also highly variable. Also nally, what we most care about is the number of deaths, rather than number of cases.
Linear fall of H(t), ln(H(t)): The original plot of H(t) vs t for Hubei can be found in [1]. It was observed that H(t) falls linearly with t, after about day-55 of the rst Covid death. Later, in [2], it was observed that linear fall of ln(H(t)) is a better t. Note that since ln(1 + E) E for small E, to a rst degree of approximation, either H(t) or ln(H(t)) would do: we observed this uniformly in all our data plots.

Data Analysis Plots for European Countries
Before plotting the data for India, we seek to understand a few essential char-acteristics of H(t) vs t for other regions where Covid-19 has progressed further than in India. We rst show the plots for UK and Sweden, where Covid-19 is showing signs of slowing but is still a signi cant concern: Fig. 1(a) shows the relevant plots.
We see that in both UK and Sweden, H(t) has fallen signi cantly over time. Like in the Hubei plot in [1], there is a good linear t to the data points, between day-55 and when H(t) has reached a value of about 1.01 (near day 100). The gure shows both linear ts: both have high R 2 values.
A common concern in the context of H(t) is that such a metric would approach 1 with increasing t, even if the number of deaths were to stay constant over time, i.e. H(t) will simply be (t + 1)/t which tends to 1 for large t. In response to this concern, we show a plot of (t + 1)/t for comparison, in Fig. 1(a). We see that throughout, the slope of H(t) is signi cantly steeper than the (t + 1)/t plot. This is consistent with the empirical observation that the number of Covid-19 deaths does reduce over time.
Finally, we observe another aspect: we also plotted ln(H(t)) vs t. However, given how close H(t) is to 1, ln(H(t)) + 1 H(t). And hence almost the same linear t carries over for ln(H(t) vs t, albeit with an offset of 1. For clarity, we have avoided showing ln(H(t)) altogether in this gure and also in subsequent plots below.
We next plot H(t) for Italy and Spain: two countries where the disease is well past its peak. Fig. 1(b) shows these plots. In both cases, H(t) of 1.01 is reached around day 75; the linear t shown is for data points between day 45 and day 75. Here too, we see a linear trend, with high R 2 . Looking across the four countries in Fig. 1, it seems likely that we can predict the H(t) in other regions too, based on the observed historical trend.
Plots for India, its States Fig. 2(a) shows the H(t) plot for India as well as the linear t for the same. We note that the R 2 value here is low. This could be because of non-uniform policies across various states after lockdown-4. However, despite the low R 2 value, we note that the linear t has good predictive power: Fig. 2(a) also shows the linear t, had we done the analysis 15-days back (i.e without the data for the most recent 15 days). We see that the linear t is almost the same in both cases: with/without the most recent 15-days data. Fig. 2(a) also shows the H(t) plot and linear t for Brazil's data. Here we see that the slope is noticeably steeper for Brazil; this merits a causal analysis, but is beyond the scope of this paper.
Next we plot H(t) for two states: Maharashtra and Gujarat, in Fig. 2(b). We note that in the case of Maharashtra, like in the case of India, the R 2 value is signi cantly low. The reason behind this is likely the fact that the majority of deaths in Maharashtra so far is from Mumbai. Even still, it is pleasantly surprising to see that the linear t has good predictive power anyway: compare the linear ts with/without the last 15 days of data and observe that they are almost the same. In the case of Gujarat, the linear t has already reached 1, suggesting that the pandemic has "ended". This compares well with ground reality, with the entire state reporting only 10-20 deaths per day since more than one week. For comparison, the expected number of deaths for Gujarat's population (6.3 crores) during "normal" times, as per 2019 death rate (7.3 per 1000 per year) is about 1260. So Covid-19 has already fallen to about just 1% of the overall cause of deaths in Gujarat.

Plots for hotspot cities in India
We next look at the three hotspot cities of Mumbai, Chennai, and Delhi. Fig. 3 shows the plots for each of the cities. In each case, we show H(t), the linear t with/without the last 15 days of data. In the case of Mumbai, we see that the linear t before 15 days predicted an earlier end, than what is predicted today. This is because the rate of slowing has lessened. This is not uncommon, and can be observed even in the cases of Italy and Spain, on closer look: the slope of H(t) becomes less steep after reaching about 1.01. So the end is "in sight", even if a bit further than earlier predicted. In the case of Chennai too, such a difference between with/without last 15 days' data is seen. And here too, H(t) has fallen to below 1.01, and a gradual lessening of the H(t) slope is to be expected. In the case of Delhi, the linear t 15-days back is quite close to the linear t now.
In all three hotspot cities, the H(t) plot suggests that the "end" of the pandemic is near. This matches with ground truth reports too [3,4].

Implications
The implications of the above analysis are many.
1. The analysis is very simple: it does not need sophisticated data scientists, only 11th/12th standard mathematic background and spreadsheet soft-ware. It is thus easily accessible to policy making 2. The linear extrapolation based predictions appear fairly effective, at least until H(t) hits low values like 01.
3. The linear trend is seen irrespective of size of region (city, state, or country), as well as irrespective of 4. The slope itself could depend on the policy (e.g. strictness of lockdown), but the linear decline of H(t) toward 1 in all regions shows clearly what every epidemiologist knows: that the pandemic "ends" through herd 5. Even in places where H(t) suggests that the pandemic is over, we can use the corresponding H(t) plots from regions where the pandemic has long ended (e.g. Italy, Spain), to convince ourselves that likely a similar pattern will occur in other regions where the pandemic has not ended. This can replace baseless fears of repeat outbreaks, with guarded 6. Extending the linear t for India as a whole, the prediction is that the pandemic will end in about 2 months from now, or toward the end of 7. Extending the linear t for Maharashtra, the prediction is that the pan-demic will end in about 5 weeks from now, or toward the rst week of Sep.
8. The linear t for the hotspot cities as well as Gujarat suggests that the pandemic is almost ended, or about a week away (end of Jul). This matches well with the ground

Conclusion
In this paper, we applied Levitt's H(t) metric on India data, and showed that its linear t has good predictive power. This has a lot of implications for data-based prediction and corresponding policy making, both in cities/states where the pandemic has almost ended, and also in cities, states, districts, and countries where it is yet to run its course.