Linear Regression and Further Analysis of Covid-19

We now notice that the way in which Covid-19 is spreading no longer fits the exponential model – which is great news as that means that we have likely avoided the worst-case scenario. In this article, we will now move forward trying to analyze the most likely scenario using linear regression to predict the spread of Covid-19 at the national and state level. We had published 3 scenarios for each state. We are now tracking at the most likely model which we had already published on our website.

By Ayesha Rajan, Research Analyst for Altheia Predictive Health

Linear and exponential regression are similar but note that while the equation used for exponential regression is y=ab^x, the equation used for linear regression is y=a+bx. Based on our findings in the last article, we will start our linear regression at or near the date at which the total number of cases falls away from the exponential curve.

Data, Analysis and Discussion

Below is a graph using exponential regression from our last article:

Total Covid-19 Cases in the United States March 4th- 26th (Exponential Regression)

From this, we can see that the number of cases begins to fall away from the exponential curve around day 25 which was March 26^th, 2020. That is the day we will label as Day 1 in this article. If we started from the same start point (March 2^nd, 2020) as this graph then we would see skewed results that take the initial exponential nature into account; so, here is the new table of data we are working with:

Total Number of Covid-19 Cases in the United States

Day	Date	Total Number of Cases (y)
1	March 25^th, 2020	68,211
2	March 26^th, 2020	85,435
3	March 27^th, 2020	104,126
4	March 28^th, 2020	123, 578
5	March 29^th, 2020	143, 491
6	March 30^th, 2020	163, 788
7	March 31^st, 2020	188, 530
8	April 1^st, 2020	215, 003
9	April 2^nd, 2020	244, 877
10	April 3^rd, 2020	277, 161
11	April 4^th, 2020	311, 357
12	April 5^th, 2020	336,673
13	April 6^th, 2020	367,004
14	April 7^th, 2020	400, 335
15	April 8^th, 2020	434, 927

Here is the resulting graph:

From this data, we have that a = 19,255.7429 and b= 26,463.83214. Though it may look like points jump over and under the line a bit, we can see a much tighter fit to the curve and also have a strong correlation coefficient at r = 0.994506795. From this, we can model out the next week as follows:

Day	Date	Predicted Total Cases (y)
17	April 10^th, 2020	410,249
18	April 11^th, 2020	436,096
19	April 12^th, 2020	461,943
20	April 13^th, 2020	487,790
21	April 14^th, 2020	513,637
22	April 15^th, 2020	539,483
23	April 16^th, 2020	565,330

We can see that compared against last week’s prediction and actual case numbers these predictions seem much more realistic. If we analyze at the state level, we can see that some states are still on the initial exponential curve while others have become more linear as well.

In our last article we looked New York, Alabama, Colorado, Missouri, Louisiana and Oregon. Of these states, New York, Louisiana, Colorado and Oregon had a close fit to an exponential curve, though New York seemed to move away from the curve at or around March 24th. The data from Alabama and Missouri did not fit the exponential curve as well as the states mentioned.

Total Covid-19 Cases in New York March 28^th– April 7^th

For New York’s linear regression equation, we have y=11,980.97143+8,308.503571x.

Total Covid-19 Cases in Louisiana March 28^th– April 7^th

For Louisiana’s linear regression equation, we have y=1,239.06667+1,507.987879x.

Total Covid-19 Cases in Alabama March 29^th– April 7^th

y=539.66667+153.715515x

Total Covid-19 Cases in Missouri March 29^th– April 7^th

y=606.2+237.8x

Total Covid-19 Cases in Colorado March 29th- April 7th

y=487.06667+64.22424242x

Total Covid-19 Cases in Colorado March 29th- April 7th

y=1,689.26667+351.2242424x

Recall from the last article the Imperial College Study that we used to predict demand for hospital beds. In the study, there were three possible scenarios based on levels of precautions taken: Optimistic Case, Most Likely Case and Worst Case. From the linear regression calculations done here, we can try to see which scenario we are leaning towards given the current situation and data. We will only predict up to June 1^st as, according to the Imperial College Model, that is the latest peak we would see amongst all scenarios.

Predicted Total Covid-19 Cases in Six Example States

	May (1^st)	Mid – May (15^th)	June (1^st)
New York	336,012	452,331	593, 576
Louisiana	54,018	75, 130	100,766
Alabama	5, 765	7,918	10,531
Missouri	8,691	12,020	16,063
Oregon	2,670	3,569	4,661
Colorado	13, 630	18, 548	24, 518

From STAT News, 5% of the total cases will need to be hospitalized so we can update the table as follows. Taking that into account, we need to look at the difference in the number of total cases between each month so we can see the number of new cases and then take 5% of that number to get the following table:

Predicted Increase in Demand for Hospital Beds Based on Linear Regression in Six Example States

	April 15^th– May 1^st	May 1^st– 15^th	May 15^th – June 1^st
New York	6,646	5,815	7,062
Louisiana	829	1055	1,281
Alabama	122	107	130
Missouri	190	166	202
Oregon	50	45	55
Colorado	281	245	299

The delta change in demand for hospital beds in New York from May to June in Imperial College’s best case scenario is 15,838. In the worst case scenario it is 32,667. According to our own calculations, it is 19,523. As more data comes in, this could change but as of right now, it seems that we are falling somewhere between Imperial College’s best and most likely scenarios. This holds true for the rest of the states we have looked at as well with some even falling below the best case scenario.

Conclusion
From our own calculations, it seems clear that social distancing and closure precautions are working – the fact that number of cases is falling away from the exponential curve is proof of that. However, despite the fact that the number of cases and demand for hospital beds is optimistic, that does not ensure that certain hospitals will not be overwhelmed. Even in the best case scenario, many hospitals in smaller counties will still fall short of demand for hospital beds.

Citations
Begley, Sharon. “Coronavirus Model Shows Individual Hospitals What to Expect.” STAT, 16 Mar. 2020, www.statnews.com/2020/03/16/coronavirus-model-shows-hospitals-what-to-expect/.