key: cord-0954740-kx5iwu9r
authors: Eng, G.
title: Initial Model for USA CoVID-19 Resurgence
date: 2020-09-21
journal: nan
DOI: 10.1101/2020.09.16.20196063
sha: 62a5047f9805362009c159a90d9373b17f60ca26
doc_id: 954740
cord_uid: kx5iwu9r

Early CoVID-19 growth obeys: N{t*}=NI exp[+Ko t* ], with Ko=[(ln2)/(tdbl)], where tdbl is the pandemic growth doubling time. Given N{t*}, the daily number of new CoVID-19 cases is {rho}{t*} = dN{t*}/d{t*}. Implementing society-wide Social Distancing increases the tdbl doubling time, and a linear function of time for tdbl was used in our Initial Model: No[t] = 1 exp[+KA t / (1 + {gamma}o t) ] = eGo exp(-Zo[t] ) , to describe these changes, where the [t]-axis is time-shifted from the t*-axis, back to the pandemic start, and Go = [ KA / {gamma}o ]. While this No[t] successfully modeled the USA CoVID-19 progress from 3/2020 to 6/2020, this equation could not easily model some quickly decreasing {rho}[t] cases ("fast pandemic shutoff"), indicating that a second process was involved. This situation was most evident in the initial CoVID-19 data from China, South Korea, and Italy. Modifying Zo[t] to allow exponential cutoffs: Zo[t] {equiv} +[Go / (1+{gamma}o t)] [exp(-{delta}ot)] = Zo[t] exp(-{delta}ot) , NA[t] = eGo exp(-ZA[t]) , resulted in an Enhanced Initial Model (EIM) that significantly improved data fits for these cases. After 6/2020, many regions of the USA "opened up", loosening their Social Distancing requirements, which led to a sudden USA CoVID-19 Resurgence. Extrapolating the USA No[t] 3/2020-6/2020 results to 9/2020 as an Initial Model Baseline (IMB), and subtracting this IMB from the newer USA data gives a Resurgence Only function, which is analyzed here. This USA CoVID-19 Resurgence function differs significantly from the No[t] IMB functional form, but it was well-modeled by the NA[t] fast pandemic shutoff function. These results indicate that: (a) the gradual increase in tdbl doubling time from society-wide shut-downs is likely due to eliminating of a large number of population gathering points that could have enabled CoVID-19 spread; and (b) having a non-zero {delta}o fast pandemic shutoff is likely due to more people wearing masks more often [with 12 Figures].

The CoVID-19 pandemic started late in 2019, becoming world-wide in early 2020, with CoVID-19 spread evolving di¤erently in various areas. Many publicly available databases were set up to track the disease, to assist epidemiologists, scientists, and policy makers in visualizing CoVID-19 spread. The widely available bing.com 1 CoVID-19 database was used here. These databases underpin model projections, allowing quick evaluation of how di¤erent inputs a¤ect the predicted outcome. Our goal was to empirically model a wide range of data with a small number of parameters, where di¤erent values for these parameters could span the range of observed CoVID-19 evolution among regions.

The N f b tg number of CoVID-19 cases starts with an exponential growth: [1.1c] where N I at time b t = 0 is the number of infected people, K o is a rate constant for how fast an infected person spreads CoVID-19 to others, and t dbl is the pandemic doubling time. This is the basis for a large number of SEIR (Susceptible, Exposed, Infected, and Recovered or Removed ) pandemic models, which are often implemented as systems of local di¤erential equations.

Implementing society-wide measures for non-infected people is inherently a non-local process. How it impacts pandemic spread is often not the main focus of SEIR models, which are local. However, when governments mandated Social Distancing, starting with shut-down of large-scale gathering places at some b t = 0 point, the N f b tg response was fairly quick. Within days, the t dbl was empirically observed to gradually lengthen, likely due to these shut-downs preventing a large number of people from gathering together and spreading CoVID-19.

Our prior work 2 3 showed that an Initial Model 2 , with a linear function of time for gradual t dbl changes:

successfully …t a lot of early CoVID-19 pandemic data. More importantly, Eq. [1.2b] showed that this Initial Model allows for CoVID-19 pandemic shut-o¤, prior to infecting the whole population.

An exception was CoVID-19 spread in Italy, having a much faster pandemic shut-o¤ than Eq. [1.2a] predicted. We attributed this to a second CoVID-19 mitigation process that was modeled with a o exponential decay time constant: 3] where o = 0 is the absence of this second process. Including this second process 4 gave an Enhanced Initial Model (EIM), which then successfully modeled CoVID-19 spread in Italy.

While the Initial Model 2 3 successfully predicted the USA CoVID-19 evolution from March 2020 through early June 2020, widespread "opening-up" of various gathering places (such as local bars and hair and nail salons) in mid-June 2020 created a large-scale USA CoVID-19 Resurgence.

A new model for USA CoVID-19 Resurgence is developed here. Our prior (3/2020-6/2020) USA CoVID-19 function was used as an Initial Model Baseline (IMB). This IMB was projected out to 9/2020, and subtracted from all followon USA data, to give a Resurgence Only function. As detailed next, the number of USA CoVID-19 Resurgence cases can substantially exceed the expected pre-Resurgence total. More importantly, this CoVID-19 Resurgence was also found to require a o 6 = 0 EIM in order to achieve a good data …t.

This o 6 = 0 result is similar to the prior analysis of CoVID-19 spread in Italy 4 . The fact that the o 6 = 0 EIM function is needed to model Resurgence, instead of an Eq. [1.2a] IMB -type function, helps to identify the CoVID-19 second process. After the Social Distancing period of 3/20-6/20, new post-6/20 society-wide recommendations or mandates to wear masks were put in place, which likely gives rise to this faster Resurgence pandemic shut-o¤.

Let N f b tg model the total number N data f b tg of CoVID-19 cases in a locality, with N data f b tg having end-points fN I ; N F g. Then N f b t = 0g = N I , and:

where t dbl is the pandemic doubling time, but if society-wide Social Distancing starts at b t = 0, then t dbl can lengthen for b t > 0. The prior b t < 0 exponential growth phase, before Social Distancing started, is not applicable for estimating Social Distancing parameters.

Our Initial Model for CoVID-19 spread and t dbl lengthening 2 3 is given in the above Eqs. points. However, all these time axes can be shifted to a new t = 0 point that estimates the CoVID-19 pandemic start:

3] which then individually sets ft I ; t F g as follows:

[2.4b] with the fN I ; t I ; N F ; t F g group uniquely determining fK A ; o g:

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The copyright holder for this this version posted September 21, 2020. 

6b] The f0 < t < t I g period, prior to Social Distancing start, estimates what the pandemic would have looked like, had Social Distancing begun at t = 0.

Using USA CoVID-19 data from bing.com 1 from 3/21/2020 through 6/7/2020, we derived the following Initial Model Baseline (IMB) best …t as shown in Figs 

7] Shortly after 6/7/2020, many states and cities around the USA "opened up" nearly simultaneously, loosening Social Distancing restrictions. This optimistic action led to a sudden USA CoVID-19 Resurgence.

To model CoVID-19 Resurgence, the Figs. 1-2 IMB curve values were subtracted from the new USA data totals. When the total number of CoVID-19 Resurgence cases, N data f b t 0 g, showed a trend above the IMB baseline, then fN I ; N F g could be used as the N data f b t 0 g data end-points. Let N f b t 0 g model this CoVID-19 Resurgence data, so that N f b t 0 = 0g = N I . Then:

and t dbl is the pandemic Resurgence doubling time. Using an Initial Resurgence Model (IRM) that parallels the prior section IMB gives:

The best fK o ; S g are set by minimizing the rms error between the Eq. 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) preprint

The copyright holder for this this version posted September 21, 2020. .

Resurgence parameters. As Eq. [3.2b] shows, this IRM allows pandemic shut-o¤ before the CoVID-19 Resurgence infects the whole population. In Eq. [3.2a] , di¤erent N I values alter the b t 0 = 0 points, but all these time axes can be shifted to a new t 0 = 0 point that estimates the CoVID-19 Resurgence start: 

at pandemic end, and the long-term tail for o [t 0 ] are each given by: Since the observed o [t 0 ] Resurgence data decreases much quicker than the IRM prediction, the USA CoVID-19 Resurgence has a fast pandemic shuto¤, which is similar to our prior study 3 4 of Italy CoVID-19 data. That Italy data was most successfully modeled by introducing a second process having an exponential decay in time. Generalizing the IRM model of Eq. [3.3a] similarly gives this Enhanced Initial Model (EIM) for Resurgence, where o 6 = 0 in Eq. [4.1c] characterizes this second process: 

[4.2e] For easier data …tting when o 6 = 0, the Eq. [4.2b] condition that N A (t 0 = 0) = 1 can be relaxed. Adjusting N A (t 0 = 0) allows N A (t 0 = t I ) = N I to be preserved. Then both ft I ; N I g can be treated as model inputs. The G o prefactor in Eq. [4.1a] can be modi…ed to give:

3] so that h A can adjust N A (t 0 = 0), while keeping the same t 0 = 0 point: 

[4.5b] The N A (t 0 ) of Eq. [4.3] then gives this A (t 0 ):

for the daily number of new CoVID-19 cases, providing a self-consistent analytic function for A (t 0 ), instead using A (t 0 ) N A (t 0 ) = t 0 as a numerical approximation. For long times, Eq. [4.6] becomes:

[4.7] which exhibits a nearly exponentially decaying tail. Minimizing the rms error using a Logarithmic Y-axis vs linear-time axis gives Figs. 5-6, with these best …t parameter values and results:

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The copyright holder for this this version posted September 21, 2020. . Fig. 6 Logarithmic Y-axis data …t is quite good, as is Fig. 5 when compared to the IRM Fig. 3 . The faster decaying Fig. 5 A (t 0 ) tail gives a signi…cantly lower prediction for the total number of Resurgence cases at the pandemic end. Finally, a similar data …t is shown in Figs. 7-8, except the rms error was minimized using a Linear Y-axis vs linear-time axis for the A (t 0 ) Resurgence data. It has the following best …t parameters, which are similar to the above Eq. [4. 8] Fig. 11 plots the ratio of the total number of deaths versus total number of cases (% vs time), based on the bing.com database 1 , which gives~2:9325% = (169; 108) = (5; 766; 718), as of 8/27/2020. This value is similar to the IHME 8/27/2020 value 5 of~3:1065%, which is shown as a horizontal line on Fig. 11 .

Using the slightly higher IHME mortality rate allows our Fig. 10 predictions   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. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 21, 2020. .

to be compared with the most recent IHME predictions 5 , as shown in Fig. 12 . The IHME predictions include the presumption of a 2 nd Resurgence, due to factors 6 of "seasonality and declining vigilance". Each IHME projection shown in Fig. 12 is also an IHME Model average 5 , with the magnitude of their lower and upper bound deviations (not graphed) being < 2:5% by 9/26/2020, increasing to < 42% by 1/1/2021. The IHME 2 nd Resurgence assumptions are evident in the upward (+) curvature in all IHME predictions, as compared to the downward (-) curvature of the present Resurgence model, indicating progress to a CoVID-19 pandemic shut-o¤, assuming NO 2 nd Resurgence occurs.

The causes of a 2 nd Resurgence could include a large-scale set of new reopenings, creating another rapid rise in CoVID-19 cases, similar to Fig. 9 . A follow-on analysis would be needed for this 2 nd Resurgence. The possibility of multiple CoVID-19 waves was highlighted early on by the University of Minnesota CoVID-19 team 7 8 , but each wave was assumed to have minimal overlap. Instead, these results, and the IHME projections (which already includes a 2 nd Resurgence), support the idea that USA CoVID-19 evolution is likely to have multiple overlapping waves of Resurgence. [3. 6b] has the o parameter accounting for the e¤ects of society-wide Social Distancing. Our prior work 2 showed that the e¤ects of implementing society-wide shut-downs changed the CoVID-19 pandemic evolution within days of the start of its implementation. Thus, the size of o likely re ‡ects the degree to which society-wide large gatherings were eliminated. It is a non-local parameter that is generally not part of the traditional SEIR (Susceptible, Exposed, Infection, and Recovered or Removed ) pandemic modeling, which are governed by local di¤erential equations.

Our analysis shows that the USA CoVID-19 Resurgence data decreased faster than the IRM model predictions. A similar situation 3 4 was seen in the CoVID-19 pandemic evolution in Italy, which was successfully modeled by introducing a second process:

[5.1] which has an exponentially decaying tail. This second process is independent of the gradually changing t dbl doubling time, which gave rise to the IRM fK A ; o g parameters.

For USA CoVID-19 Resurgence, an Enhanced Resurgence Model (ERM) was developed to include this second process. This ERM essentially replaces the

[5.2b] The necessity of using a second process ( o 6 = 0) to model the USA CoVID-19 Resurgence has a potentially important implication. This o is a second non-local parameter that may not be part of a traditional SEIR (Susceptible,

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The copyright holder for this this version posted September 21, 2020. Projections vs data 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. Figure 3 . Initial Model …t for CoVID-19 Resurgence: USA Data 6/13/20 to 8/8/20. The Fig. 1 IMB has data through 6/7/2020. It was extrapolated to 8/27/2020, then subtracted from the actual data to create Resurgence Only data, which was then …tted using the Initial Model. Figure 4 . USA CoVID-19 Resurgence Only: Data vs Initial Model Fit, 6/13/20 to 8/8/20. Initial Model Baseline (IMB) was subtracted from actual data to set Resurgence Only Data. Resurgence Start Day #1 was set to 6/13/2020 with N = 15; 650 cases above IMB. function with exponential decay term. Deviations on Logarithmic Y-axis due to minimizing error using Linear Y-axis as given in Fig. 7 . 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. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 21, 2020. . . 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) preprint

The copyright holder for this this version posted September 21, 2020. . Fig. 1 IMB has data through 6/7/2020. It was extrapolated to 8/27/2020, then subtracted from The actual data to create Resurgence Only data, which was then fitted using the Initial Model. 

Net Percent USA CoVID-19 Deaths: Ratio of Total # of USA Deaths to Total # of bing.com reported USA CoVID-19 Cases, 3/21/20 through 8/27/2020. Some USA CoVID-19 restrictions lifted~6/7/2020 leading to CoVID Resurgence; IHME used 3:1065% as of 8/27/20. Figure 12. Net Predicted USA CoVID-19 Deaths: The 8/27/20 IHME 3:1065% Mortality Rate for USA CoVID-19 cases was applied to Fig. 10 to estimate USA CoVID-19 mortality, assuming NO 2nd Resurgence; enabling comparison to IHME's model

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Projections vs data 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 fit.