key: cord-0769479-1exyjhj0
authors: Eng, G.
title: Orthogonal Functions for Evaluating Social Distancing Impact on CoVID-19 Spread
date: 2020-07-03
journal: nan
DOI: 10.1101/2020.06.30.20143149
sha: 9b9c272794d3d9321ad32344042245258dedf1b6
doc_id: 769479
cord_uid: 1exyjhj0

Early CoVID-19 growth often obeys: N{t}=N/Iexp[+K/ot], with K/o=[(ln2)/(t/dbl)], where t/dbl is the pandemic doubling time, prior to society-wide Social Distancing. Previously, we modeled Social Distancing with t/dbl as a linear function of time, where N[t]=1exp[+K/A t/(1+ gamma/o t)] is used here. Additional parameters besides {K/o,gamma/o} are needed to better model different rho[t]=dN[t]/dt shapes. Thus, a new Orthogonal Function Model [OFM] is developed here using these orthogonal function series: N(Z) = sum[m=0,M/F] g/m L/m(Z) exp[-Z] , R(Z) = sum[m=0,M/F] c/m L/m(Z) exp[-Z] , where N(Z) and Z[t] form an implicit N[t]=N(Z[t]) function, giving: G/o = [K/A / gamma/o ] , Z[t] = +[ G/o / (1+ gamma/ot) ] , rho[t] = [ gamma/o / G/o ] (Z^2) R(Z) , with L/m(Z) being the Laguerre Polynomials. At large M/F values, nearly arbitrary functions for N[t] and rho[t]=dN[t]/dt can be accommodated. How to determine {K/A, gamma/o} and the {g/m; m=(0,+M/F)} constants from any given N(Z) dataset is derived, with rho[t] set by: c/(M/F - k) = sum[m=0,k] g/m . The bing.com USA CoVID-19 data was analyzed using M/F=(0,1,2) in the OFM. All results agreed to within about 10 percent, showing model robustness. Averaging over all these predictions gives the following overall estimates for the number of USA CoVID-19 cases at the pandemic end: = 5,009,677 (+/-) 269,450 (data to 5/3/20), and = 4,422,803 (+/-) 162,580 (data to 6/7/20), which compares the pre- and post-early May bing.com revisions. The CoVID-19 pandemic in Italy was examined next. The M/F=2 limit was inadequate to model the Italy rho[t] pandemic tail. Thus, regions with a quick CoVID-19 pandemic shutoff may have additional Social Distancing factors operating, beyond what can be easily modeled by just progressively lengthening pandemic doubling times (with 13 Figures).

gm .

The bing.com USA CoVID-19 data was analyzed using MF = (0; 1; 2) in the OFM. All results agreed to within about 10 percent, showing model robustness. Averaging over all these predictions gives the following overall estimates for the number of USA CoVID-19 cases at the pandemic end:

< Nmax > = 5; 009; 677 269; 450 (data to 5/3/20), and < Nmax > = 4; 422; 803 162; 580 (data to 6/7/20), which compares the pre-and post-early May bing.com revisions. The CoVID-19 pandemic in Italy was examined next. The MF = 2 limit was inadequate to model the Italy [t] pandemic tail. Thus, regions with a quick CoVID-19 pandemic shuto¤ may have additional Social Distancing factors operating, beyond what can be easily modeled by just progressively lengthening pandemic doubling times (with 13 Figures).

The early stages of the CoVID-19 coronavirus pandemic around the world showed a nearly exponential rise in the number of infections with time. If a signi…cant fraction of the population gets infected ("saturated pandemic"), exponential growth no longer applies. However, Social Distancing can also mitigate exponential growth, enabling pandemic shuto¤ with only a small fraction of the population being infected ("dilute pandemic"). Let N f b tg be the total number of CoVID-19 cases in any given region, with f b tg being the predicted number of daily new CoVID-19 cases, so that:

[1.1a]

[1.1b] On 3/25/2020, the Institute of Health Metrics and Evaluation, University of Washington (IHME) released their initial model for CoVID-19 spread 1 where:

"The cumulative death rate for each location is assumed to follow a parametrized Gaussian error function." Since the IHME f b tg used Gaussians, their projections assumed that the rise to the pandemic peak and its subsequent fall would be symmetric. Their implicit assumption was that the amount of Social Distancing was exactly what was needed to make their model predictions true. Given a sharp f b tg rise, our concern was that the IHME model did not allow f b tg to decrease gradually.

As a result, we developed an alternative CoVID-19 spread model, which assumed 2 Social Distancing gradually lengthens the CoVID-19 doubling time. The initial exponential growth factor K o = [(ln 2)=t dbl ] was used as a starting point, where t dbl was the initial doubling time. A new Social Mitigation Parameter [SMP] S was introduced to account for society-wide Social Distancing measures. A linear function was used for doubling time lengthening as a simple extension beyond a constant K o , giving:

2b] as an Initial Model 2 for the number of CoVID-19 cases, where b t = 0 was the start of society-wide Social Distancing. Both N I = N f b t = 0g and N F = N f b t = t data end g, as the most recently available data, were treated as …xed points. A minimum root-mean-square (rms) error data…t, using a logarithmic Y-axis, sets the fK o ; S g values. The resulting N f b t ! 1g of Eq. [1.2b] is the predicted …nal number of CoVID-19 cases at the pandemic end.

On 4/29/2020, we sent our preprint 2 to the IHME, the Los Angeles Department of Public Health (LADPH), and to Profs. Goldenfeld and Maslov at UOI (University of Illinois at Urbana-Champaign), who were preparing a 2-day nationwide CoVID-19 remote-learning seminar for 5/6/2020 and 5/8/2020. Also, on 4/29/2020, the IHME electronically published their 12 th CoVID-19 update, using their 3/25/2020 model. A graphic display of their most recent f b tg projections showed a symmetric rise and fall. This graph was widely publicized by Dr. Alan Boyle, who was following the IHME work, summarizing it for general audiences 3 5 . Since our Eqs. [1.2a]-[1.2b] model gave substantially di¤erent f b tg predictions than IHME, we added a note to that e¤ect in our pre-print, submitting the …nal pre-print to MedRxiv on 5/4/2020, where it was accepted and published on-line on 5/8/2020.

Concurrently, on 5/4/2020, IHME published their 13 th CoVID-19 update 6 , where everything changed. Dr. Alan Boyle 5 summarized those changes with a note that: "[IHME] researchers acknowledged that their previous modeling wasn't sophisticated enough". Both IHME graphical predictions for 4/29/2020 and 5/4/2020 are shown in Figure 1 , to highlight this change.

On 5/6/2020 and 5/8/2020, Profs. Goldenfeld and Maslov presented their UOI team's supercomputer-based f b tg CoVID-19 projections, which also were very asymmetric. Although mathematical details for the UOI and new IHME projections are not known, virtually all f b tg CoVID-19 projections are now asymmetric, as the developing CoVID-19 data also appears to be.

Since our Initial Model had only two data …tting parameters fK o ; S g, we became concerned that those two parameters might not be su¢ cient to adequately describe all the di¤erent f b tg shapes observed. 

The following items and methods were developed as part of this OFM to improve CoVID-19 projections for a variety of f b tg data shapes.

First, the b t = 0 point in Eq. [1.2a] was time-shifted so that N [t = 0] 1. This t = 0 point now provides an estimate for the CoVID-19 pandemic starting point, replacing the above N f b tg with this time-shifted version:

which enables Eq. [2.1a] to become a 1-term approximation for a larger function series. Actual data provides the fN I ; N F g values. However, these 

It gives these equivalences between and Eq. [2.2] and Eq. [2.1f]:

[2.4c] Fourth, to allow additional data …tting parameters, the OFM replaces the 1-term approximation of Eq. [2.1a] with these orthogonal function series: is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 3, 2020. .

[2.6f] Some important properties of the Laguerre Polynomials are: 

as a new predicted total number of CoVID-19 cases at pandemic end. Eighth, the fg m ; m = (0; M F )g constants can be arranged in a ! g vector form, with comparable constants for R(Z) from Eq. [2.2] arranged in a ! C vector form. It allows Eq. [2.9] to be written as:

Once the fg m ; m = (0; M F )g constants are found, the c 0 value in Eq. [2.11] becomes the fM F + 1g-term replacement value for the predicted total number of CoVID-19 cases at the pandemic end, which re…nes the initial N o max value of Eq. [1.2b] or Eq. [2.1d]. How to determine fK A ; G o ; o g and the fg m ; m = (0; M F )g constants in Eq. [2.5a] from a given set of data, is derived next.

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For a given dataset, the OFM begins with using Eq. [1.2a] to set fK o ; S g, as in our Initial Model. Society-wide Social Distancing is assumed to occur at or before the time t I , where N I cases are already observed. Since the most recently available data at t F has N F cases, Eq. [2.1a] becomes: 

[ 

[3.4b] as separable equations to …rst …nd o , then K A , with these results: 

When data for N data (Z) are given over the whole Z = f0 + ; 1 g range, the g n constants for Eq. [2.5a] are exactly determined via:

where the Laguerre Polynomial orthogonality condition of Eq. [2.7a] forces the Eq. [4.1b] sum to reduce to one term. 6 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . When the N data (Z) only spans a …nite range of: t I < t < t F and Z min < Z < Z max , an extrapolation of N data (Z) for (Z < Z min ) and (Z > Z max ) is needed. One method could set N data (Z < Z min ) 0 and N data (Z > Z max ) 0, which results in these Eqs. [4.1a]-[4.1b] cognates:

Its advantages are: (a) for m 6 = n, every b g m and b g n are independent, as in orthogonal functions; and (b) these b g m values provide new estimates for the N data (Z < Z min ) and N data (Z > Z max ) regimes. But since N data (Z < Z min ) and N data (Z > Z max ) were originally assumed to vanish, this method is inconsistent. Alternatively, adding reasonable "tails" to the data could extend the original N data (Z) domain, but those functions are not always known.

The third path, used here, takes the Eq. [4.1a] "…nal answer " as a selfconsistent extrapolation for (Z < Z min ) and (Z > Z max ), while retaining the N data (Z) values for the (Z max Z Z min ) regime. It replaces Eq. [4.1b] with:

The fg m ; m = (0; M F )g now appears on both sides of each Eq. [4.3a] g n -equation, which is handled as follows. De…ning:

Eqs. [4.3a]-[4.3b] can be re-written as a 3 3 matrix M 3 , which relates a data-driven ! Q 3 -vector to a resultant ! g 3 -vector:

When fZ min ; Z max g ! f0; +1g, this M 3 becomes the Identity Matrix. The following k m;n (Z) integrals set K m;n :

K m;n k m;n (Z min ) k m;n (Z max ) = K n;m .

[4.6b] 7 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . The k m;n (Z) integrals can be determined using Eq. [2.7c], which gives:

[4.7f] To extract fg 0 ; g 1 ; g 2 g, the 3 3 symmetric M 3 matrix needs inversion:

which determines fg 0 ; g 1 ; g 2 g from the fQ 0 ; Q 1 ; Q 2 g data. A best …t N (Z) for Z = f0 + ; 1 g results, along with an equivalent …t for R(Z) using Eq. [2.9].

Instead of having to …nd the best fg 0 ; g 1 ; g 2 g triplet, one could …nd the best fg 0 0 ; g 0 1 g by just using using fQ 0 ; Q 1 g and an M 2 sub-matrix; or one could …nd the best fg + 0 g by itself by just using fQ 0 g and an M 1 sub-matrix:

[4.9f] When the N data (Z) is comprised of j = f1; 2; :::Jg discrete values between fZ min ; Z max g, with each Z j having an N (j) data (Z j ) value, the Eq. [4.4a] integral needs to be replaced by a sum. Let: is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . This problem was originally solved by G. N. Watson 7 in 1938, and simpli…ed by J. Gillis and G. Weiss 8 in 1960. It is a sum of terms, where each coe¢ cient contains four di¤erent factorials involving integers. Their key result is:

[4.12d] where ALL terms in the sum for n = f0; (r + s)g also have an implicit requirement that none of the integer arguments for any of the factorials can be negative. Thus, all terms with negative arguments for the factorial must be omitted. Nowadays, this calculation can be done on a computer, but it would have been di¢ cult in 1960, and nearly impossible in 1938.

This USA analysis only uses data after mid-March 2020, when several State Governors instituted mandatory Mitigation Measures. The widely available bing.com CoVID-19 data 9 for the USA had these limits: 

[5.2c] Using Eq. [3.3b] for t I and t F sets:

[5.3c] for use in the OFM. Figures 2-3 show how this Initial Model, by itself, compares to the USA CoVID-19 data. Figure 2 uses a logarithmic Y-axis for the predicted total number of CoVID-19 cases, and Figure 3 shows the daily new CoVID-19 case predictions on a linear Y-axis plot.

The daily new case data exhibits large day-to-day variations, likely due to reporting delays, among other factors. This Initial Model for the USA has a predicted maximum of~31; 760 new cases per day at Day 37:686 on 4/17/2020, along with~6; 757 new cases per day still occurring at Day 200 on 9/26/2020.

The time axis in Figure 2 is di¤erent than in our previous paper 2 , due to the time shift of Eq. [2.1a], where the new t = 0 point estimates the CoVID-19 pandemic starting point being on 3/10/2020. Even if Social Distancing had been in e¤ect at the start of the pandemic, Figure 2 shows that the N I [t I ] = f25; 722g level still could have been reached in 10 11 days. 9 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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Figures 3 compares the measured data for the total number of CoVID-19 cases after Social Distancing started, to the early-time portion of this Initial Model. That comparison shows that the early-time data starts o¤ a little below the curve; the later-time data rises a bit above the curve; and the …nal-time data again matches the curve, since it is a …xed point for this analysis.

These predictions assume: (I) The present Mitigation Measures are continued; ( Figure 4 , using a linear Y-axis, along with the t > t I raw data for the daily new CoVID-19 cases.

Raw data for t < t I was not included in these analyses, because they cover the exponential rise period, prior to Social Distancing. Those data are not applicable to estimating Social Distancing e¤ects.

However, the Figure 4 OFM provides an extrapolation for those t < t I times, which shows what an exponential rise plus lengthening doubling times would have looked like, if both had been operating continuously from the CoVID-19 pandemic start. The companion N [t] analytic result, plotted using a logarithmic Y-axis, along with the t > t I raw data for the total number of CoVID-19 cases, is show in Figure 5 .

Comparing the size and timing of the [t] pandemic peak, and its Day 200 value, between the Initial Model (Figs. 2-3) and OFM (Figs. 4-5 While the above analysis used M F = 2 with Eq. [5.10] ! g 3 setting the best fg 0 ; g 1 ; g 2 g values, the OFM also provides estimates for the simpler M F = f0; 1g cases, as outlined by Eqs. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . parameters is increased from 0 to 3. The average and 1 standard deviation among these N [t ! 1] projections is:

< N max > = 5; 009; 677 269; 450 , [5.16] where 1 is~5:4% of the overall average. Comparing the results among Figs. 2-5 highlights several items: (a) All [t] functions have a sharp rise, and a much slower decreasing tail. (b) The overall …t-to-data, as given in Fig. 3 and Fig. 5 , shows that the extra parameters in the OFM can …t the [t] shape better.

(c) The OFM helps to estimate the uncertainty in the Initial Model, which Eq. [5.16] showed was~5:4%.

(d) These results, taken together, exhibit only a relatively small change in the N [t ! 1] limits. Thus, the Initial Model function captures much of the progression to pandemic shuto¤.

The [t] tail may still di¤er from these predictions, due to factors such as: (i) The CoVID-19 dynamics may change in the long-term low [t] regime; (ii) A "second wave" or multiple waves of [t] rise and fall may occur; both of which are beyond the scope of this CoVID-19 pandemic modeling;

(iii) Using just an exponential rise at the CoVID-19 pandemic start, plus lengthening doubling times, may limit how much mitigation can be easily modeled using only a few adjustable parameters. Figure 4 provides some evidence for the above (iii) possibility. While lengthening the doubling time enables pandemic shuto¤ in the long time dilute pandemic limit; Figure 4 also shows that this model tends to approach …nal pandemic shuto¤ rather slowly.

This analysis of the bing.com USA data begins at mid-March 2020, when mandatory Mitigation Measures were instituted. However, in early-May, bing.com changed their entire database, revising all numerical values back to the start of their reporting history.

The revised bing.com USA data from mid-March through early-June is analyzed next, which had these values: is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . N [t] values, where the revised bing.com data changes were larger. Thus, some of the~10:456% change in N 1 max may be due to the revised bing.com data, but the longer (t 0 F t 0 I ) data interval also contributes to modifying the fK 0 o ; 0 S g values. The Initial Model data…t for the revised USA data is shown in Figures  6-7 , and is a better data…t than the Initial Model results of Figures 2-3 . Comparing the OFM result of Eq. [5.12], which gave N [t ! 1] = f4; 645; 874g, to the Initial Model result of Eq. [6.2c] shows that they di¤er by just~3:25%.

Next, the OFM is applied to further re…ne this Initial Model prediction. Those results are shown in Figure 8 and Figure 9 , which were derived as follows. First, the Eqs. [2.1a]-[2.1d] time-shift was done: [6.7] The ! Q 3 vector for this dataset gives this updated ! g 3 vector: 13 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . Both the Initial Model and the OFM found a comparable amount of change between the two datasets; likely due to the revised bing.com values being lower, along with the larger dataset enabling increased modeling precision.

The Initial Model and the OFM also provide self-consistent CoVID-19 predictions over the two di¤erent time periods. Each model held its predictive power to within < 10% for over a month f43 days vs: 78 daysg, without needing recalculations or parameter value changes, which provides a strong data-driven validation of the potential utility of these models. When the Initial Model is a somewhat good …t, this Orthogonal Function Model provides even better …ts.

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This Italy analysis uses data beginning on Feb. 23, 2020, from the revised bing.com CoVID-19 database 9 , which has these values:

1b] with (t F t I ) = 113 days. The number of daily new CoVID-19 cases shows a sharp post-peak decrease for Italy, in contrast the the above USA data. That sharp decrease provides a near-worst case test for the OFM. The Initial Model best …t on a logarithmic Y-axis, gives these initial parameters:

2b] Using Eq. [3.3b] for t I and t F gives:

3c] for use in the OFM. The revised bing.com Italy data and the Initial Model data…t are shown in Figure 10 and its inset. The Initial Model is not a good …t due to the high curvature of the data on the logarithmic Y-axis, which is similar to our previous 2 results for Italy. The OFM is applied next.

Using Eqs. [7.7b] The resultant symmetric matrix M 3 of K m;n entries is: 

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The copyright holder for this preprint this version posted July 3, 2020. The coe¢ cients for R(Z), which set the predicted number of daily new CoVID-19 cases for the OFM, are given by: Figure 11 , using Eqs. [2.4b] and [2.5b]. For this fast pandemic shuto¤ case, the OFM improvement over the Initial Model is not large. When the initial [exp( Z)] function is not a good …t, which is likely for quicker pandemic shuto¤s, a lot of terms, beyond the M F = 2 value used here, are needed in Eq.

[2.5a] for a good …t. An alternative method for choosing the initial [exp( Z)] function is examined next, to see if additional improvements result for that case.

There is a wide latitude in the choice of an initial [exp( Z)] function for the Eqs.

[2.5a]-[2.5b] orthogonal function expansions. However, when the Initial Model is not a good …t, the common practice of minimizing rms error using a logarithmic Y-axis for the Initial Model may not be optimal, since the Orthogonal Function Model [OFM] creates best …ts using a linear Y-axis.

Minimizing the rms error between the Initial Model and data using a linear Y-axis is done to provide an alternative [exp( Z)] function. This alternative starting point gives these parameter values, replacing Eqs. . Using Eq. [3.3b] for t I and t F gives:

2c] for use in the OFM, while still using this linear Y-axis initial …t. Figure  12 and is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. [8.4] which matches Eq. The coe¢ cients for R(Z), which set the predicted number of daily new CoVID-19 cases for the OFM, are given by: .

The Initial Model shapes for [t] were very di¤erent, depending on whether that initial data…t was performed by minimizing rms error using a logarithmic Y-axis ( Figure 10 ) or a linear Y-axis ( Figure 12 , Initial Model Re-do) as expected. However, comparing the two OFM (Figure 11 vs Figure 13 ) calculations, shows that their overall [t] shapes are quite similar.

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The copyright holder for this preprint this version posted July 3, 2020. . https://doi.org/10.1101/2020.06.30.20143149 doi: medRxiv preprint While the maxf [t p ]g calculated pandemic peaks generally increase, they are all below the data near-peak values of~4; 800 6; 500 cases/day shown in Figs. 10-13. Thus, for quick pandemic shuto¤s, the Initial Model [exp( Z)] function is less important than needing more M F terms. When the Initial Model is not a good …t, the OFM only gives limited improvements for M F = 2.

The early stages of the CoVID-19 coronavirus pandemic began with a nearly exponential rise in the number of infections with time. Let N [t] be the total number of CoVID-19 cases vs time. Our Initial Model 2 used this basic function:

[9.1b] to model Social Distancing e¤ects by progressively lengthening the doubling time for the pandemic growth. The o = 0 limit of Eq. [9.1a] corresponds to a purely exponential rise. This Initial Model enables calculation of a pandemic shuto¤ with only a small fraction of the total population becoming infected ("dilute pandemic").

To allow more data …tting parameters than just fK A ; o g, an Orthogonal Function Model [OFM] was developed, using these orthogonal function series: The expected number of daily new CoVID-19 cases, [t], is given by:

[9.3b] For a wide range of N (Z) data, larger M F and more fL m (Z); m = (0; +M F )g terms gives progressively better matches to almost any arbitrary function, enabling improved data …tting for a variety of N [t] and [t] shapes.

Methods are developed here to derive fK A ; o g, and determine the fg m ; m = (0; +M F )g and fc m ; m = (0; +M F )g constants from any given N [t] dataset. Whereas our Initial Model was an M F = 0 case, the M F = 2 case was used here for data analysis, as an OFM example.

These methods were applied to the CoVID-19 pandemic data for the USA. Analysis results using the original bing.com up data to~5/3/2020 are given in 18 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 3, 2020. . During early-May, bing.com revised their entire database, all the way back to their earliest values. This revised USA bing.com data, which included an extended time period into June 2020, was also analyzed, with results given in Figures 6-9 and Eq. [6.11].

For the USA, the Initial Model and OFM results di¤ered by only~10%, showing that the Initial Model was a somewhat good …t, while the OFM is a better …t. Comparing our calculations using the 43-day 5/3/2020 original bing.com dataset to the 78-day 6/7/2020 revised bing.com dataset, showed that our early-May USA projections predicted the June data to within . 10% for the same model. Thus, both models provided self-consistent CoVID-19 projections, holding their predictive power for over a month f43 days vs: 78 daysg, without recalculations or parameter value changes.

The Italy CoVID-19 pandemic data was studied next, as a worst-case test of the OFM. The post-May 2020 revised bing.com database was used, with results presented in Figures 10-13 Overall, both the Initial Model and this Orthogonal Function Model show how progressively lengthening the pandemic doubling time enables CoVID-19 pandemic shuto¤, even in the dilute pandemic limit. However, there may a natural limit to how fast this one mitigation factor can achieve pandemic shuto¤. For cases like Italy, other Social Distancing factors may be operating that enable and enhance quick CoVID-19 pandemic shuto¤, which are not e¤ectively being modeled by just lengthening the pandemic doubling times.

10 List of Figures   Figure 1: Comparison of IHME CoVID-19 Projections, 29 April 2020 vs 4 May 2020. CDC CoVID-19 Website highlighted IHME Projections prior to the IHME May 2020 update. Figure 2 : Initial Model for USA CoVID-19 Projections using data up to 5/3/2020. Predicted Number of Daily CoVID-19 Cases has a peak of 31,760 cases/day on 4/17; with 5,024,900 cases total; and~6,757 new cases/day at Day 200 on 9/26/2020. Figure 3 : Initial Model for USA CoVID-19 Projections vs data up to 5/3/2020. Original bing.com data up to 5/3/2020 are shown, prior to their new reporting method. Data starts slightly below, then goes slightly above the Initial Model prediction line.

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The copyright holder for this preprint this version posted July 3, 2020. . Figure 4 : Orthogonal Function Model, USA CoVID-19 Projections, data to 5/3/2020. Predicted Number of Daily CoVID-19 Cases has a peak of 32,069 cases/day on 4/15; with 4,645,874 cases total; and~5,962 new cases/day at Day 200 on 9/26/2020. Figure 5 : Orthogonal Function Model, USA CoVID-19 Projections, data to 5/3/2020. Original bing.com data up to 5/3/2020 are shown, prior to their new reporting method. Orthogonal Function Model matches data a bit better than the Initial Model. Figure 6 : Initial Model for USA CoVID-19 Projections vs data up to 6/7/2020. Predicted Number of Daily CoVID-19 Cases has a peak of 30,727 cases/day on 4/15; with 4,499,494 cases total; and~5,783 new cases/day at Day 200 on 9/27/2020. Figure 7 : Initial Model for USA CoVID-19 Projections vs data up to 6/7/2020. Revised bing.com data, circa 5/3/2020, changed all values back to the pandemic start. Initial Model appears to be a good data…t by itself. Figure 8 : Orthogonal Function Model, USA CoVID-19 Projections, data to 6/7/2020. Revised bing.com data; daily# of CoVID-19 Cases Peak at 30,909 cases/day on 4/13/2020; with 4,179,205 cases total; and~5,140 new cases/day at Day 200 on 9/27/2020. Figure 9 : Orthogonal Function Model, USA CoVID-19 Projections, data to 6/7/2020. Revised bing.com data, posted circa 5/3/2020, changed values back to the pandemic start. Orthogonal Function Model matches the data a bit better than the Initial Model. Figure 10 : Initial Model for Italy CoVID-19 Projections vs data up to 6/15/2020. Initial Model matches Total Number of Cases at data start and data end, but best …t using a logarithmic Y-axis does not give a good …t for Predicted Number of Daily CoVID-19 cases. Figure 11 : Orthogonal Function Model for Italy CoVID-19 data up to 6/15/2020. Orthogonal Function Model gives improved data…t, but 3-terms in orthogonal function series is insu¢ cient to accurately predict a rapidly decreasing Number of Daily CoVID-19 cases. Figure 12 : Initial Model re-do, Italy CoVID-19 data to 6/15/2020. New starting point is a best …t function on a linear Y-axis, instead of having a best …t using a logarithmic Y-axis. Alternative method may allow a few-term orthogonal function series to better match the data. Figure 13 : Orthogonal Function Model re-do, Italy CoVID-19 data to 6/15/2020. Orthogonal Function Model re-do gives a slightly better smallseries …t. Other Social Distancing impacts likely exist besides just lengthening pandemic doubling times.

https://www.MedRxiv.org/ ID=MedRxiv/2020/043752v1, 03.25.2020, "Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator-days and deaths by US state in the next 4 months", 20 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 3, 2020. . https://doi.org/10.1101/2020.06.30.20143149 doi: medRxiv preprint 

Coronavirus modelers raise projected U.S. death toll and lengthen state-by-state recovery timeline

A Note of the Polynomials of Hermite and Laguerre

Products of Laguerre Polynomials